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+ + + + L + + + c + + + + + + + diff --git a/doc/equations/beta2.svg b/doc/equations/beta2.svg new file mode 100644 index 000000000..8eebc13d5 --- /dev/null +++ b/doc/equations/beta2.svg @@ -0,0 +1,2 @@ + +beta(a,b)=(a+g−0.5a+b+g−0.5)a−0.5(b+g−0.5a+b+g−0.5)beb+g−0.5L(a)L(b)L(c) \ No newline at end of file diff --git a/doc/equations/beta3.mml b/doc/equations/beta3.mml new file mode 100644 index 000000000..30b7935c5 --- /dev/null +++ b/doc/equations/beta3.mml @@ -0,0 +1,128 @@ + + + beta + + + a + , + b + + + = + + + + + + a + + + g + − + 0.5 + + + a + + + b + + + g + − + 0.5 + + + + + + a + − + b + − + 0.5 + + + + + + + + + + a + + + g + − + 0.5 + + + + + b + + + g + − + 0.5 + + + + + + + + a + + + b + + + g + − + 0.5 + + + 2 + + + + + + b + + + + + e + + b + + + g + − + 0.5 + + + + + + + L + + + a + + + L + + + b + + + + + L + + + c + + + + + + + diff --git a/doc/equations/beta3.svg b/doc/equations/beta3.svg new file mode 100644 index 000000000..73250d4c8 --- /dev/null +++ b/doc/equations/beta3.svg @@ -0,0 +1,2 @@ + +beta(a,b)=(a+g−0.5a+b+g−0.5)a−b−0.5((a+g−0.5)(b+g−0.5)(a+b+g−0.5)2)beb+g−0.5L(a)L(b)L(c) \ No newline at end of file diff --git a/doc/equations/beta4.mml b/doc/equations/beta4.mml new file mode 100644 index 000000000..d3fe984fd --- /dev/null +++ b/doc/equations/beta4.mml @@ -0,0 +1,34 @@ + + + beta + + + a + , + b + + + = + + + + a + + + b + + + b + + beta + + + a + , + b + + + 1 + + + + + diff --git a/doc/equations/beta4.svg b/doc/equations/beta4.svg new file mode 100644 index 000000000..f130637eb --- /dev/null +++ b/doc/equations/beta4.svg @@ -0,0 +1,2 @@ + +beta(a,b)=(a+b)bbeta(a,b+1) \ No newline at end of file diff --git a/doc/equations/beta5.mml b/doc/equations/beta5.mml new file mode 100644 index 000000000..bf290a38b --- /dev/null +++ b/doc/equations/beta5.mml @@ -0,0 +1,49 @@ + + + beta + + + a + , + b + + + = + + + + + a + + + b + + + + + a + + + b + + + 1 + + + + + a + b + + + beta + + + a + + + 1, + b + + + 1 + + + + + diff --git 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beta + + + a + , + b + + + = + + 1 + b + + + + diff --git a/doc/equations/beta7.svg b/doc/equations/beta7.svg new file mode 100644 index 000000000..ee8feaf66 --- /dev/null +++ b/doc/equations/beta7.svg @@ -0,0 +1,2 @@ + +ifa=1then:beta(a,b)=1b \ No newline at end of file diff --git a/doc/equations/beta8.mml b/doc/equations/beta8.mml new file mode 100644 index 000000000..288ef1244 --- /dev/null +++ b/doc/equations/beta8.mml @@ -0,0 +1,52 @@ + + + + e + + l + c + − + l + a + − + l + b + + + + + + + + l + a + + + l + c + + + + + a + + + + + + + l + b + + + l + c + + + + + b + + + + diff --git a/doc/equations/beta8.svg b/doc/equations/beta8.svg new file mode 100644 index 000000000..a059630be --- /dev/null +++ b/doc/equations/beta8.svg @@ -0,0 +1,2 @@ + +elc−la−lb(lalc)a(lblc)b \ No newline at end of file diff --git a/doc/equations/beta_dist_kurtosis.mml b/doc/equations/beta_dist_kurtosis.mml new file mode 100644 index 000000000..42e2a32b1 --- /dev/null +++ b/doc/equations/beta_dist_kurtosis.mml @@ -0,0 +1,72 @@ + + + 6 + + + + α + 3 + + − + + α + 2 + + + + 2 + β + − + 1 + + + + + + β + 2 + + + + β + − + 1 + + + − + 2 + α + β + + + β + + + 2 + + + + + α + β + + + α + + + β + + + 2 + + + + + α + + + β + + + 3 + + + + + + + diff --git a/doc/equations/beta_dist_kurtosis.svg b/doc/equations/beta_dist_kurtosis.svg new file mode 100644 index 000000000..66d0fa779 --- /dev/null +++ b/doc/equations/beta_dist_kurtosis.svg @@ -0,0 +1,2 @@ + +6α3−α2(2β−1)+β2(β−1)−2αβ(β+2)αβ(α+β+2)(α+β+3) \ No newline at end of file diff --git a/doc/equations/binomial_ref1.mml b/doc/equations/binomial_ref1.mml new file mode 100644 index 000000000..6b334c385 --- /dev/null +++ b/doc/equations/binomial_ref1.mml @@ -0,0 +1,212 @@ + + + + + + f + + + k + ; + n + , + p + + + + + = + + + + + n + + + C + k + + + p + k + + + + + 1 + − + p + + + + n + − + k + + + + + + + + + + = + + + + + n + ! + + + k + ! + + + n + − + k + + + ! + + + + p + k + + + + + 1 + − + p + + + + n + − + k + + + + + + + + + + = + + + + + Γ + + + n + + + 1 + + + + + Γ + + + k + + + 1 + + + Γ + + + n + − + k + + + 1 + + + + + + p + k + + + + + 1 + − + p + + + + n + − + k + + + + + + + + + + = + + + + + + p + k + + + + + 1 + − + p + + + + n + − + k + + + + + B + + + k + + + 1, + n + − + k + + + 1 + + + + + n + + + 1 + + + + + + + + + + diff --git a/doc/equations/binomial_ref1.svg b/doc/equations/binomial_ref1.svg new file mode 100644 index 000000000..ec7cef8f8 --- /dev/null +++ b/doc/equations/binomial_ref1.svg @@ -0,0 +1,2 @@ + +f(k;n,p)=nCkpk(1−p)n−k=n!k!(n−k)!pk(1−p)n−k=Γ(n+1)Γ(k+1)Γ(n−k+1)pk(1−p)n−k=pk(1−p)n−kB(k+1,n−k+1)(n+1) \ No newline at end of file diff --git a/doc/equations/binomial_ref2.mml b/doc/equations/binomial_ref2.mml new file mode 100644 index 000000000..4d660be95 --- /dev/null +++ b/doc/equations/binomial_ref2.mml @@ -0,0 +1,98 @@ + + + + + + f + + + k + ; + n + , + p + + + + + = + + + + + n + + + C + k + + + p + k + + + + + 1 + − + p + + + + n + − + k + + + + + + + + + + = + + + + + n + ! + + + k + ! + + + n + − + k + + + ! + + + + p + k + + + + + 1 + − + p + + + + n + − + k + + + + + + + + diff --git a/doc/equations/binomial_ref2.svg b/doc/equations/binomial_ref2.svg new file mode 100644 index 000000000..25b7e9938 --- /dev/null +++ b/doc/equations/binomial_ref2.svg @@ -0,0 +1,2 @@ + +f(k;n,p)=nCkpk(1−p)n−k=n!k!(n−k)!pk(1−p)n−k \ No newline at end of file diff --git a/doc/equations/cauchy_ref1.mml b/doc/equations/cauchy_ref1.mml new file mode 100644 index 000000000..a8c68ca72 --- /dev/null +++ b/doc/equations/cauchy_ref1.mml @@ -0,0 +1,51 @@ + + + f + + + x + ; + + x + 0, + + γ + + + + = + + + 1 + π + + + + + γ + + + + + x + − + + x + 0 + + + + 2 + + + + + γ + 2 + + + + + + + + diff --git a/doc/equations/cauchy_ref1.svg b/doc/equations/cauchy_ref1.svg new file mode 100644 index 000000000..ee6c6a8a7 --- /dev/null +++ b/doc/equations/cauchy_ref1.svg @@ -0,0 +1,2 @@ + +f(x;x0,γ)=1π(γ(x−x0)2+γ2) \ No newline at end of file diff --git a/doc/equations/chf.mml b/doc/equations/chf.mml new file mode 100644 index 000000000..b848bbe9e --- /dev/null +++ b/doc/equations/chf.mml @@ -0,0 +1,44 @@ + + + chf + + + d + i + s + t + , + x + + + + = + + H + + + x + + + + = + + + ∫ + + − + ∞ + + x + + h + + + μ + + + d + μ + + + diff --git a/doc/equations/chf.svg b/doc/equations/chf.svg new file mode 100644 index 000000000..b2e59a7a4 --- /dev/null +++ b/doc/equations/chf.svg @@ -0,0 +1,2 @@ + +chf(dist,x)=H(x)=∫−∞xh(μ)dμ \ No newline at end of file diff --git a/doc/equations/chi_squ_ref1.mml b/doc/equations/chi_squ_ref1.mml new file mode 100644 index 000000000..d10a98cf9 --- /dev/null +++ b/doc/equations/chi_squ_ref1.mml @@ -0,0 +1,44 @@ + + + Q + + = + + + ∑ + + i + = + 1 + + ν + + + + + + + + χ + i + + − + + ν + i + + + + + σ + i + + + + + + 2 + + + + diff --git a/doc/equations/chi_squ_ref1.svg b/doc/equations/chi_squ_ref1.svg new file mode 100644 index 000000000..bf94addcc --- /dev/null +++ b/doc/equations/chi_squ_ref1.svg @@ -0,0 +1,2 @@ + +Q=∑i=1ν(χi−νiσi)2 \ No newline at end of file diff --git a/doc/equations/chi_squ_tut1.mml b/doc/equations/chi_squ_tut1.mml new file mode 100644 index 000000000..2c3790831 --- /dev/null +++ b/doc/equations/chi_squ_tut1.mml @@ -0,0 +1,90 @@ + + + + + + + + + N + − + 1 + + + + s + 2 + + + + + χ + + + + + α + 2, + + N + − + 1 + + + + 2 + + + + + + + ≤ + + σ + + ≤ + + + + + + + + N + − + 1 + + + + s + 2 + + + + + χ + + + + 1 + − + + α + 2, + + N + − + 1 + + + + 2 + + + + + + + + diff --git a/doc/equations/chi_squ_tut1.svg b/doc/equations/chi_squ_tut1.svg new file mode 100644 index 000000000..ee6be2210 --- /dev/null +++ b/doc/equations/chi_squ_tut1.svg @@ -0,0 +1,2 @@ + +(N−1)s2χ(α2,N−1)2≤σ≤(N−1)s2χ(1−α2,N−1)2 \ No newline at end of file diff --git a/doc/equations/chi_squ_tut2.mml b/doc/equations/chi_squ_tut2.mml new file mode 100644 index 000000000..166e6f835 --- /dev/null +++ b/doc/equations/chi_squ_tut2.mml @@ -0,0 +1,24 @@ + + + + + χ + + + + + α + 2, + + N + − + 1 + + + + 2 + + + + + diff --git a/doc/equations/chi_squ_tut2.svg b/doc/equations/chi_squ_tut2.svg new file mode 100644 index 000000000..03475f0e6 --- /dev/null +++ b/doc/equations/chi_squ_tut2.svg @@ -0,0 +1,2 @@ + +χ(α2,N−1)2 \ No newline at end of file diff --git a/doc/equations/chi_squ_tut3.mml b/doc/equations/chi_squ_tut3.mml new file mode 100644 index 000000000..562d7a253 --- /dev/null +++ b/doc/equations/chi_squ_tut3.mml @@ -0,0 +1,26 @@ + + + + + χ + + + + 1 + − + + α + 2, + + N + − + 1 + + + + 2 + + + + + diff --git a/doc/equations/chi_squ_tut3.svg b/doc/equations/chi_squ_tut3.svg new file mode 100644 index 000000000..aa10b98a0 --- /dev/null +++ b/doc/equations/chi_squ_tut3.svg @@ -0,0 +1,2 @@ + +χ(1−α2,N−1)2 \ No newline at end of file diff --git a/doc/equations/derivative1.mml b/doc/equations/derivative1.mml new file mode 100644 index 000000000..d07677fd4 --- /dev/null +++ b/doc/equations/derivative1.mml @@ -0,0 +1,61 @@ + + + gamma_p_derivative + + + a + , + x + + + + = + + + ∂ + + ∂ + x + + + P + + + a + , + x + + + + = + + + + + e + + − + x + + + + x + + a + − + 1 + + + + + Γ + + + a + + + + + + + diff --git a/doc/equations/derivative1.svg b/doc/equations/derivative1.svg new file mode 100644 index 000000000..b976dcfb6 --- /dev/null +++ b/doc/equations/derivative1.svg @@ -0,0 +1,2 @@ + +gamma_p_derivative(a,x)=∂∂xP(a,x)=e−xxa−1Γ(a) \ No newline at end of file diff --git a/doc/equations/derivative2.mml b/doc/equations/derivative2.mml new file mode 100644 index 000000000..40f0f44d5 --- /dev/null +++ b/doc/equations/derivative2.mml @@ -0,0 +1,75 @@ + + + ibeta_derivative + + + a + , + b + , + x + + + + = + + + ∂ + + ∂ + x + + + + I + x + + + + a + , + b + + + + = + + + + + + + 1 + − + x + + + + b + − + 1 + + + + x + + a + − + 1 + + + + + B + + + a + , + b + + + + + + + diff --git a/doc/equations/derivative2.svg b/doc/equations/derivative2.svg new file mode 100644 index 000000000..53612b144 --- /dev/null +++ b/doc/equations/derivative2.svg @@ -0,0 +1,2 @@ + +ibeta_derivative(a,b,x)=∂∂xIx(a,b)=(1−x)b−1xa−1B(a,b) \ No newline at end of file diff --git a/doc/equations/digamma1.mml b/doc/equations/digamma1.mml new file mode 100644 index 000000000..13c3ef69f --- /dev/null +++ b/doc/equations/digamma1.mml @@ -0,0 +1,56 @@ + + + ψ + + + x + + + + = + + + d + + d + x + + + ln + + + Γ + + + x + + + + + + = + + + + + Γ + ′ + + + + x + + + + + Γ + + + x + + + + + + + diff --git a/doc/equations/digamma1.svg b/doc/equations/digamma1.svg new file mode 100644 index 000000000..e17a78d3c --- /dev/null +++ b/doc/equations/digamma1.svg @@ -0,0 +1,2 @@ + +ψ(x)=ddxln(Γ(x))=Γ′(x)Γ(x) \ No newline at end of file diff --git a/doc/equations/digamma2.mml b/doc/equations/digamma2.mml new file mode 100644 index 000000000..f764615df --- /dev/null +++ b/doc/equations/digamma2.mml @@ -0,0 +1,60 @@ + + + ψ + + + x + + + + = + + ln + + + x + + + + + + 1 + + 2 + x + + + − + + ∑ + + n + = + 1 + + ∞ + + + + + B + + 2 + n + + + + + 2 + n + + x + + 2 + n + + + + + + + diff --git a/doc/equations/digamma2.svg b/doc/equations/digamma2.svg new file mode 100644 index 000000000..7a77a35d5 --- /dev/null +++ b/doc/equations/digamma2.svg @@ -0,0 +1,2 @@ + +ψ(x)=ln(x)+12x−∑n=1∞B2n2nx2n \ No newline at end of file diff --git a/doc/equations/digamma3.mml b/doc/equations/digamma3.mml new file mode 100644 index 000000000..a1ca38026 --- /dev/null +++ b/doc/equations/digamma3.mml @@ -0,0 +1,94 @@ + + + ψ + + + x + + + + = + + + + z + − + + 1 + 2 + + + + x + + + g + − + + 1 + 2 + + + + + + ln + + + x + + + g + − + + 1 + 2 + + + + + + + + + P + ′ + + + + x + + + + + P + + + x + + + + + − + + + + Q + ′ + + + + x + + + + + Q + + + x + + + + + − + 1 + + + diff --git a/doc/equations/digamma3.svg b/doc/equations/digamma3.svg new file mode 100644 index 000000000..0c6cf9ff0 --- /dev/null +++ b/doc/equations/digamma3.svg @@ -0,0 +1,2 @@ + +ψ(x)=z−12x+g−12+ln(x+g−12)+P′(x)P(x)−Q′(x)Q(x)−1 \ No newline at end of file diff --git a/doc/equations/dist_tutorial1.mml b/doc/equations/dist_tutorial1.mml new file mode 100644 index 000000000..7145560ea --- /dev/null +++ b/doc/equations/dist_tutorial1.mml @@ -0,0 +1,140 @@ + + + + + + t + + = + + + + Sm + + + 1 + + − + + Sm + 2 + + + + + s + p + + + + + 1 + + S n + 1 + + + + + + 1 + + S n + 2 + + + + + + + + + + + + s + p + + + = + + + + + + + + + Sn + 1 + + − + 1 + + + + Sd + + 1 + + 2 + + + + + + + Sn + 2 + + − + 1 + + + + Sd + + 2 + + 2 + + + + + Sn + 1 + + + + + Sn + 2 + + − + 2 + + + + + + + + + ν + + = + + + Sn + 1 + + + + + Sn + 2 + + − + 2 + + + + + + diff --git a/doc/equations/dist_tutorial1.svg b/doc/equations/dist_tutorial1.svg new file mode 100644 index 000000000..748ec731e --- /dev/null +++ b/doc/equations/dist_tutorial1.svg @@ -0,0 +1,2 @@ + +t=Sm1−Sm2sp1S n1+1S n2sp=(Sn1−1)Sd12+(Sn2−1)Sd22Sn1+Sn2−2ν=Sn1+Sn2−2 \ No newline at end of file diff --git a/doc/equations/dist_tutorial2.mml b/doc/equations/dist_tutorial2.mml new file mode 100644 index 000000000..e7e3fef05 --- /dev/null +++ b/doc/equations/dist_tutorial2.mml @@ -0,0 +1,55 @@ + + + t + + = + + + + + Sm + 1 + + − + + Sm + 2 + + + + + + + + Sd + + 1 + + 2 + + + Sn + 1 + + + + + + + Sd + + 2 + + 2 + + + Sn + 2 + + + + + + + + + diff --git a/doc/equations/dist_tutorial2.svg b/doc/equations/dist_tutorial2.svg new file mode 100644 index 000000000..c2fb1f4b7 --- /dev/null +++ b/doc/equations/dist_tutorial2.svg @@ -0,0 +1,2 @@ + +t=Sm1−Sm2Sd12Sn1+Sd22Sn2 \ No newline at end of file diff --git a/doc/equations/dist_tutorial3.mml b/doc/equations/dist_tutorial3.mml new file mode 100644 index 000000000..1a5efa232 --- /dev/null +++ b/doc/equations/dist_tutorial3.mml @@ -0,0 +1,114 @@ + + + ν + + = + + + + + + + + + Sd + + 1 + + 2 + + + Sn + 1 + + + + + + + Sd + + 2 + + 2 + + + Sn + 2 + + + + + 2 + + + + + + + + + + Sd + + 1 + + 2 + + + Sn + 1 + + + + + 2 + + + + + Sn + 1 + + − + 1 + + + + + + + + + + + + Sd + + 2 + + 2 + + + Sn + 2 + + + + + 2 + + + + + Sn + 2 + + − + 1 + + + + + + + + diff --git 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+ + + x + , + − + y + + + + = + + + + + + x + + x + + + y + + + + + + + 1 + 2 + + + + + R + C + + + + x + + + y + , + y + + + + + diff --git a/doc/equations/ellint18.svg b/doc/equations/ellint18.svg new file mode 100644 index 000000000..5bd74886d --- /dev/null +++ b/doc/equations/ellint18.svg @@ -0,0 +1,2 @@ + +RC(x,−y)=(xx+y)12RC(x+y,y) \ No newline at end of file diff --git a/doc/equations/ellint19.mml b/doc/equations/ellint19.mml new file mode 100644 index 000000000..ed1a26602 --- /dev/null +++ b/doc/equations/ellint19.mml @@ -0,0 +1,119 @@ + + + + + + F + + + − + φ + , + k + + + + + = + + + − + F + + + φ + , + k + + + + + + + F + + + φ + + + m + π + , + k + + + + + = + + + F + + + φ + , + k + + + + + 2 + m + K + + + k + + + + + + + F + + + φ + , + k + + + + + = + + + sin + φ + + R + F + + + + cos + + + 2 + + φ + , + 1 + − + + k + 2 + + sin + + + 2 + + φ + , + 1 + + + + + + + + diff --git a/doc/equations/ellint19.svg b/doc/equations/ellint19.svg new file mode 100644 index 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newline at end of file diff --git a/doc/equations/gamm1.mml b/doc/equations/gamm1.mml new file mode 100644 index 000000000..d2a6842c7 --- /dev/null +++ b/doc/equations/gamm1.mml @@ -0,0 +1,43 @@ + + + tgamma + + + z + + + = + Γ + + + z + + + = + + ∫ + + 0 + + ∞ + + + t + + z + − + 1 + + + + e + + − + t + + + d + t + + + diff --git a/doc/equations/gamm1.svg b/doc/equations/gamm1.svg new file mode 100644 index 000000000..03d152b77 --- /dev/null +++ b/doc/equations/gamm1.svg @@ -0,0 +1,2 @@ + +tgamma(z)=Γ(z)=∫0∞tz−1e−tdt \ No newline at end of file diff --git a/doc/equations/gamm2.mml b/doc/equations/gamm2.mml new file mode 100644 index 000000000..b57d0dc83 --- /dev/null +++ b/doc/equations/gamm2.mml @@ -0,0 +1,130 @@ + + + Γ + + + z + + + = + + + + l + a + + + e + + − + l + + + + + + + + ∑ + + 0 + + ∞ + + + + + l + n + + + a + + + n + − + + + + + + + + + 1 + + a + − + + + c + 1 + + + + d + 1 + + + + + + c + 2 + + + + d + 2 + + + + + + c + 3 + + + + d + 3 + + + + … + + + + + + + + + + + ; + + c + k + + = + − + + + a + + + k + − + 1 + + + ∧ + + d + k + + = + a + + + k + + + l + + + diff --git a/doc/equations/gamm2.svg b/doc/equations/gamm2.svg new file mode 100644 index 000000000..850780cc6 --- /dev/null +++ b/doc/equations/gamm2.svg @@ -0,0 +1,2 @@ + +Γ(z)=(lae−l)(∑0∞lnan−+1a−c1d1+c2d2+c3d3+…);ck=−(a+k−1)∧dk=a+k+l \ No newline at end of file diff --git a/doc/equations/gamm3.mml b/doc/equations/gamm3.mml new file mode 100644 index 000000000..9d5178d50 --- /dev/null +++ b/doc/equations/gamm3.mml @@ -0,0 +1,25 @@ + + + Γ + + + z + + + = + + + Γ + + + z + + + 1 + + + + z + + + + diff --git a/doc/equations/gamm3.svg b/doc/equations/gamm3.svg new file mode 100644 index 000000000..5946143ba --- /dev/null +++ b/doc/equations/gamm3.svg @@ -0,0 +1,2 @@ + +Γ(z)=Γ(z+1)z \ No newline at end of file diff --git a/doc/equations/gamm4.mml b/doc/equations/gamm4.mml new file mode 100644 index 000000000..84b529840 --- /dev/null +++ b/doc/equations/gamm4.mml @@ -0,0 +1,28 @@ + + + + lim + + z + → + 0 + + + + + Γ + + + z + + + + + = + + 1 + z + + + + diff --git a/doc/equations/gamm4.svg b/doc/equations/gamm4.svg new file mode 100644 index 000000000..f87d45b61 --- /dev/null +++ b/doc/equations/gamm4.svg @@ -0,0 +1,2 @@ + +limz→0(Γ(z))=1z \ No newline at end of file diff --git a/doc/equations/gamm5.mml b/doc/equations/gamm5.mml new file mode 100644 index 000000000..1d3a8de95 --- /dev/null +++ b/doc/equations/gamm5.mml @@ -0,0 +1,33 @@ + + + Γ + + + − + z + + + = + − + + π + + Γ + + + z + + + sin + + + π + z + + + z + + + + + diff --git a/doc/equations/gamm5.svg b/doc/equations/gamm5.svg new file mode 100644 index 000000000..4eb369b4d --- /dev/null +++ b/doc/equations/gamm5.svg @@ -0,0 +1,2 @@ + +Γ(−z)=−πΓ(z)sin(πz)z \ No newline at end of file diff --git a/doc/equations/gamma_dist_ref1.mml b/doc/equations/gamma_dist_ref1.mml new file mode 100644 index 000000000..f67ba1ffd --- /dev/null +++ b/doc/equations/gamma_dist_ref1.mml @@ -0,0 +1,50 @@ + + + f + + + x + ; + k + , + θ + + + + = + + + x + + k + − + 1 + + + + + e + + − + + x + θ + + + + + + θ + k + + Γ + + + k + + + + + + + diff --git a/doc/equations/gamma_dist_ref1.svg b/doc/equations/gamma_dist_ref1.svg new file mode 100644 index 000000000..8ee2e0a87 --- /dev/null +++ b/doc/equations/gamma_dist_ref1.svg @@ -0,0 +1,2 @@ + +f(x;k,θ)=xk−1e−xθθkΓ(k) \ No newline at end of file diff --git a/doc/equations/gamma_dist_ref2.mml b/doc/equations/gamma_dist_ref2.mml new file mode 100644 index 000000000..d01a24ea1 --- /dev/null +++ b/doc/equations/gamma_dist_ref2.mml @@ -0,0 +1,50 @@ + + + f + + + x + ; + α + , + β + + + + = + + + x + + α + − + 1 + + + + + + β + α + + + e + + − + β + x + + + + + Γ + + + α + + + + + + + diff --git a/doc/equations/gamma_dist_ref2.svg b/doc/equations/gamma_dist_ref2.svg new file mode 100644 index 000000000..ba6c53d21 --- /dev/null +++ b/doc/equations/gamma_dist_ref2.svg @@ -0,0 +1,2 @@ + +f(x;α,β)=xα−1βαe−βxΓ(α) \ No newline at end of file diff --git a/doc/equations/gamma_ratio.xml b/doc/equations/gamma_ratio.xml index 25ae385f6..783d82ca6 100644 --- a/doc/equations/gamma_ratio.xml +++ b/doc/equations/gamma_ratio.xml @@ -9,7 +9,7 @@ - tgamma_ratio(aﾸb) + tgamma_ratio(a,b) = @@ -35,7 +35,7 @@ - tgamma_delta_ratio(aﾸdelta) + tgamma_delta_ratio(a,delta) = @@ -68,3 +68,4 @@ + diff --git a/doc/equations/gamma_ratio0.mml b/doc/equations/gamma_ratio0.mml new file mode 100644 index 000000000..2a04b16fd --- /dev/null +++ b/doc/equations/gamma_ratio0.mml @@ -0,0 +1,28 @@ + + + tgamma_ratio(a,b) + + = + + + + Γ + + + a + + + + + Γ + + + b + + + + + + + + diff --git a/doc/equations/gamma_ratio0.svg b/doc/equations/gamma_ratio0.svg new file mode 100644 index 000000000..afbd4e9f3 --- /dev/null +++ b/doc/equations/gamma_ratio0.svg @@ -0,0 +1,2 @@ + +tgamma_ratio(a,b)=Γ(a)Γ(b) \ No newline at end of file diff --git a/doc/equations/gamma_ratio1.mml b/doc/equations/gamma_ratio1.mml new file mode 100644 index 000000000..ba8645667 --- /dev/null +++ b/doc/equations/gamma_ratio1.mml @@ -0,0 +1,34 @@ + + + tgamma_delta_ratio(a,delta) + + = + + + + Γ + + + a + + + + + Γ + + + a + + + d + e + l + t + a + + + + + + + + diff --git a/doc/equations/gamma_ratio1.svg b/doc/equations/gamma_ratio1.svg new file mode 100644 index 000000000..4139afdcb --- /dev/null +++ b/doc/equations/gamma_ratio1.svg @@ -0,0 +1,2 @@ + +tgamma_delta_ratio(a,delta)=Γ(a)Γ(a+delta) \ No newline at end of file diff --git a/doc/equations/hazard.mml b/doc/equations/hazard.mml new file mode 100644 index 000000000..0b0502cb8 --- /dev/null +++ b/doc/equations/hazard.mml @@ -0,0 +1,43 @@ + + + hazard + + + x + + + + = + + h + + + x + + + + = + + + + pdf + + + x + + + + + 1 + − + cdf + + + x + + + + + + + diff --git a/doc/equations/hazard.svg b/doc/equations/hazard.svg new file mode 100644 index 000000000..80ddee434 --- /dev/null +++ b/doc/equations/hazard.svg @@ -0,0 +1,2 @@ + +hazard(x)=h(x)=pdf(x)1−cdf(x) \ No newline at end of file diff --git a/doc/equations/hermite_0.mml b/doc/equations/hermite_0.mml new file mode 100644 index 000000000..abc2439e7 --- /dev/null +++ b/doc/equations/hermite_0.mml @@ -0,0 +1,67 @@ + + + hermite + + + x + + + + = + + + H + n + + + + x + + + + = + + + + + − + 1 + + + n + + + e + + x + 2 + + + + + + d + 2 + + + + d + + x + 2 + + + + + e + + − + + x + 2 + + + + + + diff --git a/doc/equations/hermite_0.svg b/doc/equations/hermite_0.svg new file mode 100644 index 000000000..64751fb17 --- /dev/null +++ b/doc/equations/hermite_0.svg @@ -0,0 +1,2 @@ + +hermite(x)=Hn(x)=(−1)nex2d2dx2e−x2 \ No newline at end of file diff --git a/doc/equations/hermite_1.mml b/doc/equations/hermite_1.mml new file mode 100644 index 000000000..4ef3e8c6a --- /dev/null +++ b/doc/equations/hermite_1.mml @@ -0,0 +1,48 @@ + + + + H + + n + + + 1 + + + + + x + + + + = + + 2 + x + + H + n + + + + x + + + − + 2 + n + + H + + n + − + 1 + + + + + x + + + + + diff --git a/doc/equations/hermite_1.svg b/doc/equations/hermite_1.svg new file mode 100644 index 000000000..64a76f0ba --- /dev/null +++ b/doc/equations/hermite_1.svg @@ -0,0 +1,2 @@ + +Hn+1(x)=2xHn(x)−2nHn−1(x) \ No newline at end of file diff --git a/doc/equations/hypot.mml b/doc/equations/hypot.mml new file mode 100644 index 000000000..fea0cc06c --- /dev/null +++ b/doc/equations/hypot.mml @@ -0,0 +1,29 @@ + + + hypot + + + x + , + y + + + + = + + + + + x + 2 + + + + + y + 2 + + + + + + diff --git a/doc/equations/hypot.svg b/doc/equations/hypot.svg new file mode 100644 index 000000000..4c3a8acc8 --- /dev/null +++ b/doc/equations/hypot.svg @@ -0,0 +1,2 @@ + +hypot(x,y)=x2+y2 \ No newline at end of file diff --git a/doc/equations/hypot2.mml b/doc/equations/hypot2.mml new file mode 100644 index 000000000..d31f1c5d5 --- /dev/null +++ b/doc/equations/hypot2.mml @@ -0,0 +1,34 @@ + + + hypot + + + x + , + y + + + + = + + x + + + 1 + + + + + + + y + x + + + + 2 + + + + + + diff --git a/doc/equations/hypot2.svg b/doc/equations/hypot2.svg new file mode 100644 index 000000000..ccd9499be --- /dev/null +++ b/doc/equations/hypot2.svg @@ -0,0 +1,2 @@ + +hypot(x,y)=x1+(yx)2 \ No newline at end of file diff --git a/doc/equations/ibeta1.mml b/doc/equations/ibeta1.mml new file mode 100644 index 000000000..fc98174dc --- /dev/null +++ b/doc/equations/ibeta1.mml @@ -0,0 +1,63 @@ + + + beta + + + a + , + b + , + x + + + + = + + + B + x + + + + a + , + b + + + + = + + + ∫ + + 0 + + x + + + t + + a + − + 1 + + + + + + 1 + − + t + + + + b + − + 1 + + + d + t + + + diff --git a/doc/equations/ibeta1.svg b/doc/equations/ibeta1.svg new file mode 100644 index 000000000..31de12f8e --- /dev/null +++ b/doc/equations/ibeta1.svg @@ -0,0 +1,2 @@ + +beta(a,b,x)=Bx(a,b)=∫0xta−1(1−t)b−1dt \ No newline at end of file diff --git a/doc/equations/ibeta10.mml b/doc/equations/ibeta10.mml new file mode 100644 index 000000000..0aa5f5966 --- /dev/null +++ b/doc/equations/ibeta10.mml @@ -0,0 +1,202 @@ + + + + + + + I + x + + + + a + , + b + + + + + + = + + + + + I + x + + + + a + + + n + , + b + + + + + + x + a + + + + + 1 + − + x + + + b + + + ∑ + + j + = + 1 + + n + + + + Γ + + + a + + + b + + + j + − + 1 + + + + + Γ + + + b + + + Γ + + + a + + + j + + + + + + x + + j + − + 1 + + + + + + + + + = + + + + + I + x + + + + a + + + n + , + b + + + + + + + + x + a + + + + + 1 + − + x + + + b + + + a + + + ∑ + + j + = + 0 + + + n + − + 1 + + + + + + + + a + + + b + + + + + j + − + + + + + + + + + a + + + 1 + + + + + j + − + + + + + + + x + j + + + + + + + diff --git a/doc/equations/ibeta10.svg b/doc/equations/ibeta10.svg new file mode 100644 index 000000000..e644520bc --- /dev/null +++ b/doc/equations/ibeta10.svg @@ -0,0 +1,2 @@ + +Ix(a,b)=Ix(a+n,b)+xa(1−x)b∑j=1nΓ(a+b+j−1)Γ(b)Γ(a+j)xj−1=Ix(a+n,b)+xa(1−x)ba∑j=0n−1(a+b)j−(a+1)j−xj \ No newline at end of file diff --git a/doc/equations/ibeta11.mml b/doc/equations/ibeta11.mml new file mode 100644 index 000000000..c14b6c27c --- /dev/null +++ b/doc/equations/ibeta11.mml @@ -0,0 +1,135 @@ + + + + B + x + + + + a + , + b + + + + = + + + + + + + a + + + b + + + + + n + − + + + + + + + a + + + n + − + + + + + + + B + x + + + + a + + + n + , + b + + + + + + + + x + a + + + + + 1 + − + x + + + b + + + a + + + ∑ + + j + = + 0 + + + n + − + 1 + + + + + + + + a + + + b + + + + + j + − + + + + + + + + + a + + + 1 + + + + + j + − + + + + + + + x + j + + + + diff --git a/doc/equations/ibeta11.svg b/doc/equations/ibeta11.svg new file mode 100644 index 000000000..958442197 --- /dev/null +++ b/doc/equations/ibeta11.svg @@ -0,0 +1,2 @@ + +Bx(a,b)=(a+b)n−an−Bx(a+n,b)+xa(1−x)ba∑j=0n−1(a+b)j−(a+1)j−xj \ No newline at end of file diff --git a/doc/equations/ibeta12.mml b/doc/equations/ibeta12.mml new file mode 100644 index 000000000..a080245a5 --- /dev/null +++ b/doc/equations/ibeta12.mml @@ -0,0 +1,77 @@ + + + + I + x + + + + a + , + b + + + + = + + + ∑ + + i + = + k + + + 1 + + N + + + + N + ! + + + i + ! + + + N + − + i + + + ! + + + + x + i + + + y + + + N + − + i + + + + + ; + + k + = + a + − + 1, + N + = + a + + + b + − + 1 + + + diff --git a/doc/equations/ibeta12.svg b/doc/equations/ibeta12.svg new file mode 100644 index 000000000..a3eb39416 --- /dev/null +++ b/doc/equations/ibeta12.svg @@ -0,0 +1,2 @@ + +Ix(a,b)=∑i=k+1NN!i!(N−i)!xiy(N−i);k=a−1,N=a+b−1 \ No newline at end of file diff --git a/doc/equations/ibeta2.mml b/doc/equations/ibeta2.mml new file mode 100644 index 000000000..bc4124b55 --- /dev/null +++ b/doc/equations/ibeta2.mml @@ -0,0 +1,49 @@ + + + betac + + + a + , + b + , + x + + + + = + + 1 + − + + B + x + + + + a + , + b + + + + = + + + B + + 1 + − + x + + + + + b + , + a + + + + + diff --git a/doc/equations/ibeta2.svg b/doc/equations/ibeta2.svg new file mode 100644 index 000000000..f7c3fbb75 --- /dev/null +++ b/doc/equations/ibeta2.svg @@ -0,0 +1,2 @@ + +betac(a,b,x)=1−Bx(a,b)=B1−x(b,a) \ No newline at end of file diff --git a/doc/equations/ibeta3.mml b/doc/equations/ibeta3.mml new file mode 100644 index 000000000..dc7519b5c --- /dev/null +++ b/doc/equations/ibeta3.mml @@ -0,0 +1,76 @@ + + + ibeta + + + a + , + b + , + x + + + + = + + + I + x + + + + a + , + b + + + + = + + + 1 + + beta + + + a + , + b + + + + + + ∫ + + 0 + + x + + + t + + a + − + 1 + + + + + + 1 + − + t + + + + b + − + 1 + + + d + t + + + diff --git a/doc/equations/ibeta3.svg b/doc/equations/ibeta3.svg new file mode 100644 index 000000000..f23c66b39 --- /dev/null +++ b/doc/equations/ibeta3.svg @@ -0,0 +1,2 @@ + +ibeta(a,b,x)=Ix(a,b)=1beta(a,b)∫0xta−1(1−t)b−1dt \ No newline at end of file diff --git a/doc/equations/ibeta4.mml b/doc/equations/ibeta4.mml new file mode 100644 index 000000000..0c21c5d3c --- /dev/null +++ b/doc/equations/ibeta4.mml @@ -0,0 +1,49 @@ + + + ibetac + + + a + , + b + , + x + + + + = + + 1 + − + + I + x + + + + a + , + b + + + + = + + + I + + 1 + − + x + + + + + b + , + a + + + + + diff --git a/doc/equations/ibeta4.svg b/doc/equations/ibeta4.svg new file mode 100644 index 000000000..f1d9a6b88 --- /dev/null +++ b/doc/equations/ibeta4.svg @@ -0,0 +1,2 @@ + +ibetac(a,b,x)=1−Ix(a,b)=I1−x(b,a) \ No newline at end of file diff --git a/doc/equations/ibeta5.mml b/doc/equations/ibeta5.mml new file mode 100644 index 000000000..4845bd785 --- /dev/null +++ b/doc/equations/ibeta5.mml @@ -0,0 +1,185 @@ + + + + + + + I + x + + + + a + , + b + + + + = + + + + + x + a + + + y + b + + + + a + B + + + a + , + b + + + + + + + + 1 + + 1 + + + + + d + 1 + + + 1 + + + + + d + 2 + + + 1 + + + … + + + + + + + + + + + + + + + + d + + 2 + m + + + 1 + + + = + + + − + + + a + + + m + + + + + a + + + b + + + m + + + x + + + + + a + + + 2 + m + + + + + a + + + 2 + m + + + 1 + + + + + + + + + + + + d + + 2 + m + + + = + + + m + + + b + − + m + + + x + + + + + a + + + 2 + m + − + 1 + + + + + a + + + 2 + m + + + + + + + + + + diff --git a/doc/equations/ibeta5.svg b/doc/equations/ibeta5.svg new file mode 100644 index 000000000..b1c2f2be8 --- /dev/null +++ b/doc/equations/ibeta5.svg @@ -0,0 +1,2 @@ + +Ix(a,b)=xaybaB(a,b)(11+d11+d21+…)d2m+1=−(a+m)(a+b+m)x(a+2m)(a+2m+1)d2m=m(b−m)x(a+2m−1)(a+2m) \ No newline at end of file diff --git a/doc/equations/ibeta6.mml b/doc/equations/ibeta6.mml new file mode 100644 index 000000000..d698c028d --- /dev/null +++ b/doc/equations/ibeta6.mml @@ -0,0 +1,247 @@ + + + + + + + I + x + + + + a + , + b + + + + = + + + + + x + a + + + y + b + + + + B + + + + a + , + b + + + + + + + + + + α + 1 + + + + β + 1 + + + + + + α + 2 + + + + β + 2 + + + + + + α + 3 + + + + β + 3 + + + + … + + + + + + + + + + + + + + α + 1 + + = + 1 + + , + + + α + + m + + + 1 + + + = + + + + + a + + + m + − + 1 + + + + + a + + + b + + + m + − + 1 + + + m + + + b + − + m + + + + x + 2 + + + + + + + a + + + 2 + m + − + 1 + + + 2 + + + + + + + + + β + + m + + + 1 + + + = + m + + + + + m + + + b + − + m + + + x + + + a + + + 2 + m + − + 1 + + + + + + + + + a + + + m + + + + + a + − + + + a + + + b + + + x + + + 1 + + + m + + + 2 + − + x + + + + + + + a + + + 2 + m + + + 1 + + + + + + + + diff --git a/doc/equations/ibeta6.svg b/doc/equations/ibeta6.svg new file mode 100644 index 000000000..b9a933eb8 --- /dev/null +++ b/doc/equations/ibeta6.svg @@ -0,0 +1,2 @@ + +Ix(a,b)=xaybB(a,b)(α1β1+α2β2+α3β3+…)α1=1,αm+1=(a+m−1)(a+b+m−1)m(b−m)x2(a+2m−1)2βm+1=m+m(b−m)xa+2m−1+(a+m)(a−(a+b)x+1+m(2−x))a+2m+1 \ No newline at end of file diff --git a/doc/equations/ibeta7.mml b/doc/equations/ibeta7.mml new file mode 100644 index 000000000..edde64963 --- /dev/null +++ b/doc/equations/ibeta7.mml @@ -0,0 +1,80 @@ + + + + I + x + + + + a + , + b + + + + = + + + + + x + a + + + + B + + + a + , + b + + + + + + ∑ + + n + = + 1 + + ∞ + + + + + + + 1 + − + b + + + + + n + − + + + + + x + n + + + + + + a + + + n + + + n + ! + + + + + diff --git a/doc/equations/ibeta7.svg b/doc/equations/ibeta7.svg new file mode 100644 index 000000000..93f5af7ab --- /dev/null +++ b/doc/equations/ibeta7.svg @@ -0,0 +1,2 @@ + +Ix(a,b)=xaB(a,b)∑n=1∞(1−b)n−xn(a+n)n! \ No newline at end of file diff --git a/doc/equations/ibeta8.mml b/doc/equations/ibeta8.mml new file mode 100644 index 000000000..b2d9ddf37 --- /dev/null +++ b/doc/equations/ibeta8.mml @@ -0,0 +1,243 @@ + + + + + + + I + x + + + + a + , + b + + + + ≈ + + M + + ∑ + + n + = + 0 + + ∞ + + + p + n + + + J + n + + + + b + , + u + + + + ; + + a + > + b + + + + + M + = + + + H + + + b + , + u + + + Γ + + + a + + + b + + + + + Γ + + + a + + + + T + b + + + + + + + + H + + + c + , + u + + + = + + + + e + + − + u + + + + u + c + + + + Γ + + + a + + + + + + + + + T + = + a + + + + + b + − + 1 + + 2 + + + + u + = + − + T + ln + + + x + + + + + + + + p + 0 + + = + 1 + + + p + n + + = + + + + + b + − + 1 + + + + + + + 2 + n + − + 1 + + + ! + + + + + + 1 + n + + + ∑ + + m + = + 1 + + + n + − + 1 + + + + + + m + b + − + n + + + + + + 2 + m + + + 1 + + + ! + + + + p + + n + − + m + + + + + + + + diff --git a/doc/equations/ibeta8.svg b/doc/equations/ibeta8.svg new file mode 100644 index 000000000..ecfe73bea --- /dev/null +++ b/doc/equations/ibeta8.svg @@ -0,0 +1,2 @@ + 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a + − + 1 + + + + e + + − + t + + + d + t + + + diff --git a/doc/equations/igamma1.svg b/doc/equations/igamma1.svg new file mode 100644 index 000000000..fb1d1a975 --- /dev/null +++ b/doc/equations/igamma1.svg @@ -0,0 +1,2 @@ + +tgamma(a,z)=Γ(a,z)=∫z∞ta−1e−tdt \ No newline at end of file diff --git a/doc/equations/igamma10.mml b/doc/equations/igamma10.mml new file mode 100644 index 000000000..39dda6d30 --- /dev/null +++ b/doc/equations/igamma10.mml @@ -0,0 +1,57 @@ + + + γ + + + a + , + x + + + + = + + + x + a + + + ∑ + + k + = + 0 + + ∞ + + + + + + + − + 1 + + + k + + + x + k + + + + + + a + + + k + + + k + ! + + + + + diff --git a/doc/equations/igamma10.svg b/doc/equations/igamma10.svg new file mode 100644 index 000000000..104560b03 --- /dev/null +++ b/doc/equations/igamma10.svg @@ -0,0 +1,2 @@ + +γ(a,x)=xa∑k=0∞(−1)kxk(a+k)k! \ No newline at end of file diff --git a/doc/equations/igamma11.mml b/doc/equations/igamma11.mml new file mode 100644 index 000000000..d1fb928e6 --- /dev/null +++ b/doc/equations/igamma11.mml @@ -0,0 +1,124 @@ + + + + + + Γ + + + a + , + x + + + + = + + + + tgamma1pm1 + + + a + + + − + powm1 + + + x + , + a + + + + a + + + + + x + a + + + ∑ + + k + = + 1 + + ∞ + + + + + + + − + 1 + + + k + + + x + k + + + + + + a + + + k + + + k + ! + + + + + + + tgamma1pm1 + + + a + + + = + Γ + + + a + + + 1 + + + − + 1 + + + + + powm1 + + + x + , + a + + + = + + x + a + + − + 1 + + + + + + diff --git a/doc/equations/igamma11.svg b/doc/equations/igamma11.svg new file mode 100644 index 000000000..bfad5d859 --- /dev/null +++ b/doc/equations/igamma11.svg @@ -0,0 +1,2 @@ + +Γ(a,x)=tgamma1pm1(a)−powm1(x,a)a+xa∑k=1∞(−1)kxk(a+k)k!tgamma1pm1(a)=Γ(a+1)−1powm1(x,a)=xa−1 \ No newline at end of file diff --git a/doc/equations/igamma11b.mml b/doc/equations/igamma11b.mml new file mode 100644 index 000000000..369538b9c --- /dev/null +++ b/doc/equations/igamma11b.mml @@ -0,0 +1,104 @@ + + + gp1m1 + + + z + + + + = + + + + 1 + + + + + z + + g + + + 0.5 + + + + + + + + [ + + + + + + z + + + g + − + 0.5 + + e + + + + z + + − + 1 + ] + + + [ + L + + + z + + + − + 1 + ] + + + + + + z + + + g + − + 0.5 + + e + + + + z + + + + + + [ + + + 1 + + + + + z + + g + + + 0.5 + + + + + + − + 1 + ] + + + diff --git a/doc/equations/igamma11b.svg b/doc/equations/igamma11b.svg new file mode 100644 index 000000000..90bb4ec49 --- /dev/null +++ b/doc/equations/igamma11b.svg @@ -0,0 +1,2 @@ + +gp1m1(z)=1+zg+0.5([(z+g−0.5e)z−1]+[L(z)−1](z+g−0.5e)z)+[1+zg+0.5−1] \ No newline at end of file diff --git a/doc/equations/igamma12.mml b/doc/equations/igamma12.mml new file mode 100644 index 000000000..058caaf6d --- /dev/null +++ b/doc/equations/igamma12.mml @@ -0,0 +1,81 @@ + + + + + + x + a + + + e + + − + x + + + + + Γ + + + a + + + + + + = + + + e + + x + − + a + + + + + + + x + + a + + + g + − + 0.5 + + + + + a + + + + + + a + + + g + − + 0.5 + + e + + + + + 1 + + L + + + a + + + + + + + diff --git a/doc/equations/igamma12.svg b/doc/equations/igamma12.svg new file mode 100644 index 000000000..8b6393414 --- /dev/null +++ b/doc/equations/igamma12.svg @@ -0,0 +1,2 @@ + +xae−xΓ(a)=ex−a(xa+g−0.5)aa+g−0.5e1L(a) \ No newline at end of file diff --git a/doc/equations/igamma13.mml b/doc/equations/igamma13.mml new file mode 100644 index 000000000..4f3b51f14 --- /dev/null +++ b/doc/equations/igamma13.mml @@ -0,0 +1,104 @@ + + + + e + + x + − + a + + + + + + + x + + a + + + g + − + 0.5 + + + + + a + + + = + + + e + + + a + log1pmx + + + + + x + − + a + − + g + + + 0.5 + + + a + + + g + − + 0.5 + + + + + + + + + x + + + 0.5 + − + g + + + + + a + + + g + − + 0.5 + + + + + + + ; + + log1pmx + + + z + + + = + ln + + + 1 + + + z + + + − + z + + + diff --git a/doc/equations/igamma13.svg b/doc/equations/igamma13.svg new file mode 100644 index 000000000..597dc67fe --- /dev/null +++ b/doc/equations/igamma13.svg @@ -0,0 +1,2 @@ + +ex−a(xa+g−0.5)a=e(alog1pmx(x−a−g+0.5a+g−0.5)+x(0.5−g)a+g−0.5);log1pmx(z)=ln(1+z)−z \ No newline at end of file diff --git a/doc/equations/igamma16.mml 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100644 index 000000000..388cb2161 --- /dev/null +++ b/doc/equations/igamma17.svg @@ -0,0 +1,2 @@ + +Q(a,x)=12erfc(y)+e−y2πaT(a,λ);λ>1 \ No newline at end of file diff --git a/doc/equations/igamma18.mml b/doc/equations/igamma18.mml new file mode 100644 index 000000000..929f54f65 --- /dev/null +++ b/doc/equations/igamma18.mml @@ -0,0 +1,57 @@ + + + λ + = + + x + a + + + , + + y + = + a + + + λ + − + 1 + − + ln + λ + + + = + − + a + + + ln + + + 1 + + + σ + + + − + σ + + + + ; + + σ + = + + + x + − + a + + a + + + + diff --git a/doc/equations/igamma18.svg b/doc/equations/igamma18.svg new file mode 100644 index 000000000..04382789c --- /dev/null +++ b/doc/equations/igamma18.svg @@ -0,0 +1,2 @@ + +λ=xa,y=a(λ−1−lnλ)=−a(ln(1+σ)−σ);σ=x−aa \ No newline at end of file diff --git a/doc/equations/igamma19.mml b/doc/equations/igamma19.mml new file mode 100644 index 000000000..433cfdb87 --- /dev/null +++ b/doc/equations/igamma19.mml @@ -0,0 +1,75 @@ + + + T + + + a + , + λ + + + + = + + + ∑ + + k + = + 0 + + N + + + + + ∑ + + n + = + 0 + + M + + + C + + k + + n + + + z + n + + + + + a + + − + k + + + + ; + + z + = + sign + + + λ + − + 1 + + + + + 2 + σ + + + + + diff --git a/doc/equations/igamma19.svg b/doc/equations/igamma19.svg new file mode 100644 index 000000000..b255960a0 --- /dev/null +++ b/doc/equations/igamma19.svg @@ -0,0 +1,2 @@ + +T(a,λ)=∑k=0N(∑n=0MCknzn)a−k;z=sign(λ−1)2σ \ No newline at end of file diff --git a/doc/equations/igamma1f.mml b/doc/equations/igamma1f.mml new file mode 100644 index 000000000..860a586ce --- /dev/null +++ b/doc/equations/igamma1f.mml @@ -0,0 +1,57 @@ + + + Q + + + a + , + x + + + + = + + + e + + − + x + + + + ∑ + + n + = + 0 + + + a + − + 1 + + + + + + x + n + + + + n + ! + + + + ; + + a + ∈ + + ℕ + + + + + + diff --git a/doc/equations/igamma1f.svg b/doc/equations/igamma1f.svg new file mode 100644 index 000000000..4516fdaa5 --- /dev/null +++ b/doc/equations/igamma1f.svg @@ -0,0 +1,2 @@ + +Q(a,x)=e−x∑n=0a−1xnn!;a∈ℕ+ \ No newline at end of file diff --git a/doc/equations/igamma2.mml b/doc/equations/igamma2.mml new file mode 100644 index 000000000..03afa3117 --- /dev/null +++ b/doc/equations/igamma2.mml @@ -0,0 +1,51 @@ + + + tgamma_lower + + + a + , + z + + + + = + + γ + + + a + , + z + + + + = + + + ∫ + + 0 + + z + + + t + + a + − + 1 + + + + e + + − + t + + + d + t + + + diff --git a/doc/equations/igamma2.svg b/doc/equations/igamma2.svg new file mode 100644 index 000000000..e064cddf1 --- /dev/null +++ b/doc/equations/igamma2.svg @@ -0,0 +1,2 @@ + +tgamma_lower(a,z)=γ(a,z)=∫0zta−1e−tdt \ No newline at end of file diff --git a/doc/equations/igamma2f.mml b/doc/equations/igamma2f.mml new file mode 100644 index 000000000..ca8da8873 --- /dev/null +++ b/doc/equations/igamma2f.mml @@ -0,0 +1,98 @@ + + + Q + + + a + , + x + + + + = + + erfc + + + + x + + + + + + + + e + + − + x + + + + + π + x + + + + + ∑ + + n + = + 1 + + i + + + + x + n + + + + + 1 + − + + 1 + 2 + + + + … + + + n + − + + 1 + 2 + + + + + + + ; + + a + = + i + + + + 1 + 2 + + + ; + + i + ∈ + + ℕ + + + + + + diff --git a/doc/equations/igamma2f.svg b/doc/equations/igamma2f.svg new file mode 100644 index 000000000..971e7cb9e --- /dev/null +++ b/doc/equations/igamma2f.svg @@ -0,0 +1,2 @@ + +Q(a,x)=erfc(x)+e−xπx∑n=1ixn(1−12)…(n−12);a=i+12;i∈ℕ+ \ No newline at end of file diff --git a/doc/equations/igamma3.mml b/doc/equations/igamma3.mml new file mode 100644 index 000000000..6824ac5b6 --- /dev/null +++ b/doc/equations/igamma3.mml @@ -0,0 +1,85 @@ + + + gamma_q + + + a + , + z + + + + = + + Q + + + a + , + z + + + + = + + + + Γ + + + a + , + z + + + + + Γ + + + a + + + + + + = + + + 1 + + Γ + + + a + + + + + + ∫ + + z + + ∞ + + + t + + a + − + 1 + + + + e + + − + t + + + d + t + + + diff --git a/doc/equations/igamma3.svg b/doc/equations/igamma3.svg new file mode 100644 index 000000000..6e240199e --- /dev/null +++ b/doc/equations/igamma3.svg @@ -0,0 +1,2 @@ + +gamma_q(a,z)=Q(a,z)=Γ(a,z)Γ(a)=1Γ(a)∫z∞ta−1e−tdt \ No newline at end of file diff --git a/doc/equations/igamma4.mml b/doc/equations/igamma4.mml new file mode 100644 index 000000000..11a32394c --- /dev/null +++ b/doc/equations/igamma4.mml @@ -0,0 +1,85 @@ + + + gamma_p + + + a + , + z + + + + = + + P + + + a + , + z + + + + = + + + + γ + + + a + , + z + + + + + Γ + + + a + + + + + + = + + + 1 + + Γ + + + a + + + + + + ∫ + + 0 + + z + + + t + + a + − + 1 + + + + e + + − + t + + + d + t + + + diff --git a/doc/equations/igamma4.svg b/doc/equations/igamma4.svg new file mode 100644 index 000000000..ac0f9d2d3 --- /dev/null +++ b/doc/equations/igamma4.svg @@ -0,0 +1,2 @@ + +gamma_p(a,z)=P(a,z)=γ(a,z)Γ(a)=1Γ(a)∫0zta−1e−tdt \ No newline at end of file diff --git a/doc/equations/igamma5.mml b/doc/equations/igamma5.mml new file mode 100644 index 000000000..6075fe042 --- /dev/null +++ b/doc/equations/igamma5.mml @@ -0,0 +1,26 @@ + + + Q + + + a + , + x + + + + + P + + + a + , + x + + + + = + + 1 + + + diff --git a/doc/equations/igamma5.svg b/doc/equations/igamma5.svg new file mode 100644 index 000000000..820f9eba4 --- /dev/null +++ b/doc/equations/igamma5.svg @@ -0,0 +1,2 @@ + +Q(a,x)+P(a,x)=1 \ No newline at end of file diff --git a/doc/equations/igamma6.mml b/doc/equations/igamma6.mml new file mode 100644 index 000000000..b617ec160 --- /dev/null +++ b/doc/equations/igamma6.mml @@ -0,0 +1,31 @@ + + + Γ + + + a + , + z + + + + + γ + + + a + , + z + + + + = + + Γ + + + a + + + + + diff --git a/doc/equations/igamma6.svg b/doc/equations/igamma6.svg new file mode 100644 index 000000000..aed303bc8 --- /dev/null +++ b/doc/equations/igamma6.svg @@ -0,0 +1,2 @@ + +Γ(a,z)+γ(a,z)=Γ(a) \ No newline at end of file diff --git a/doc/equations/igamma7.mml b/doc/equations/igamma7.mml new file mode 100644 index 000000000..e58ec2727 --- /dev/null +++ b/doc/equations/igamma7.mml @@ -0,0 +1,70 @@ + + + Q + + + a + , + z + + + + = + + + + Γ + + + a + , + z + + + + + Γ + + + a + + + + + + , + + P + + + a + , + z + + + + = + + + + γ + + + a + , + z + + + + + Γ + + + a + + + + + + + diff --git a/doc/equations/igamma7.svg b/doc/equations/igamma7.svg new file mode 100644 index 000000000..fd2f4ba78 --- /dev/null +++ b/doc/equations/igamma7.svg @@ -0,0 +1,2 @@ + +Q(a,z)=Γ(a,z)Γ(a),P(a,z)=γ(a,z)Γ(a) \ No newline at end of file diff --git a/doc/equations/igamma8.mml b/doc/equations/igamma8.mml new file mode 100644 index 000000000..b3084311b --- /dev/null +++ b/doc/equations/igamma8.mml @@ -0,0 +1,108 @@ + + + γ + + + a + , + x + + + + = + + + x + a + + + e + + − + x + + + + ∑ + + k + = + 0 + + ∞ + + + + + Γ + + + a + + + + + Γ + + + a + + + k + + + 1 + + + + + + + x + n + + + = + + + x + a + + + e + + − + x + + + + ∑ + + k + = + 0 + + ∞ + + + + + x + n + + + + + a + + + k + + + 1 + + − + + + + + + + diff --git a/doc/equations/igamma8.svg b/doc/equations/igamma8.svg new file mode 100644 index 000000000..aab96f886 --- /dev/null +++ b/doc/equations/igamma8.svg @@ -0,0 +1,2 @@ + +γ(a,x)=xae−x∑k=0∞Γ(a)Γ(a+k+1)xn=xae−x∑k=0∞xnak+1− \ No newline at end of file diff --git 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+ x + + + + = + + + + + e + x + + + + n + ! + + + + + + d + n + + + + d + + x + n + + + + + + + x + n + + + e + + − + x + + + + + + + diff --git a/doc/equations/laguerre_0.svg b/doc/equations/laguerre_0.svg new file mode 100644 index 000000000..c8491b32d --- /dev/null +++ b/doc/equations/laguerre_0.svg @@ -0,0 +1,2 @@ + +laguerre(n,x)=Ln(x)=exn!dndxn(xne−x) \ No newline at end of file diff --git a/doc/equations/laguerre_1.mml b/doc/equations/laguerre_1.mml new file mode 100644 index 000000000..830db0603 --- /dev/null +++ b/doc/equations/laguerre_1.mml @@ -0,0 +1,70 @@ + + + laguerre + + + n + , + m + , + x + + + + = + + + L + + n + + m + + + + x + + + + = + + + + + − + 1 + + + m + + + + + d + m + + + + d + + x + m + + + + + L + + n + + + m + + + + + x + + + + + diff --git a/doc/equations/laguerre_1.svg b/doc/equations/laguerre_1.svg new file mode 100644 index 000000000..37a836bf7 --- /dev/null +++ b/doc/equations/laguerre_1.svg @@ -0,0 +1,2 @@ + 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+ m + + + + x + + + − + + + m + + + n + + + n + + + 1 + + + + L + + n + − + 1 + + m + + + + x + + + + + diff --git a/doc/equations/laguerre_3.svg b/doc/equations/laguerre_3.svg new file mode 100644 index 000000000..d7c5a7d26 --- /dev/null +++ b/doc/equations/laguerre_3.svg @@ -0,0 +1,2 @@ + +Ln+1m(x)=m+2n+1−xn+1Lnm(x)−m+nn+1Ln−1m(x) \ No newline at end of file diff --git a/doc/equations/lanczos0.mml b/doc/equations/lanczos0.mml new file mode 100644 index 000000000..b35220cf9 --- /dev/null +++ b/doc/equations/lanczos0.mml @@ -0,0 +1,62 @@ + + + Γ + + + z + + + 1 + + + + = + + + + 2 + π + + + + + + z + + + g + + + 0.5 + + + + z + + + 0.5 + + + + e + + − + + + z + + + g + + + 0.5 + + + + + + S + g + + + + z + + + + + diff --git a/doc/equations/lanczos0.svg b/doc/equations/lanczos0.svg new file mode 100644 index 000000000..9c3f6598b --- /dev/null +++ b/doc/equations/lanczos0.svg @@ -0,0 +1,2 @@ + +Γ(z+1)=2π(z+g+0.5)z+0.5e−(z+g+0.5)Sg(z) \ No newline at end of file diff --git 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+ e + + + + + + diff --git a/doc/equations/lanczos3.svg b/doc/equations/lanczos3.svg new file mode 100644 index 000000000..402d143ee --- /dev/null +++ b/doc/equations/lanczos3.svg @@ -0,0 +1,2 @@ + +Dij={0ifi≠j1ifi=j=0−1ifi=j=1Di−1,i−12(2i−1)i−1otherwise \ No newline at end of file diff --git a/doc/equations/lanczos4.mml b/doc/equations/lanczos4.mml new file mode 100644 index 000000000..856aefc6d --- /dev/null +++ b/doc/equations/lanczos4.mml @@ -0,0 +1,107 @@ + + + + B + + i + , + j + + + = + { + + + + 1 + + + i + f + + + i + = + 0 + + + + + − + + 1 + + j + − + i + + + X + + + i + f + + + i + > + 0 + + + j + ≥ + i + + + + + 0 + + + + + + o + t + h + e + r + w + i + s + e + + + + + + ; + + + X + = + + + + + + i + + + j + − + 1 + + + + + j + − + i + + + + + + + + diff --git a/doc/equations/lanczos4.svg b/doc/equations/lanczos4.svg new file mode 100644 index 000000000..af79befee --- /dev/null +++ b/doc/equations/lanczos4.svg @@ -0,0 +1,2 @@ + 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@@ -0,0 +1,2 @@ + +Γ(z)=(z+g−0.5e)z−0.5Lg,e(z); \ No newline at end of file diff --git a/doc/equations/legendre_0.mml b/doc/equations/legendre_0.mml new file mode 100644 index 000000000..d45a3a4bf --- /dev/null +++ b/doc/equations/legendre_0.mml @@ -0,0 +1,75 @@ + + + legendre_p + + + l + , + x + + + + = + + + P + l + + + + x + + + + = + + + 1 + + + 2 + l + + l + ! + + + + + + d + l + + + + d + + x + l + + + + + + + + x + 2 + + − + 1 + + + l + + + ; + + | + x + | + ≤ + 1 + + + diff --git a/doc/equations/legendre_0.svg b/doc/equations/legendre_0.svg new file mode 100644 index 000000000..89f6fb14e --- /dev/null +++ b/doc/equations/legendre_0.svg @@ -0,0 +1,2 @@ + +legendre_p(l,x)=Pl(x)=12ll!dldxl(x2−1)l;|x|≤1 \ No newline at end of file diff --git a/doc/equations/legendre_1.mml b/doc/equations/legendre_1.mml new file mode 100644 index 000000000..8f6208cf9 --- /dev/null +++ b/doc/equations/legendre_1.mml @@ -0,0 +1,84 @@ + + + legendre_p + + + l + , + m + , + x + + + + = + + + P + + l + + m + + + + x + + + + = + + + + + − + 1 + + + m + + + + + 1 + − + + x + 2 + + + + + + m + 2 + + + + + + + d + m + + + P + l + + + + x + + + + + d + + x + m + + + + + + diff --git a/doc/equations/legendre_1.svg b/doc/equations/legendre_1.svg new file mode 100644 index 000000000..0d7443961 --- /dev/null +++ b/doc/equations/legendre_1.svg @@ -0,0 +1,2 @@ + +legendre_p(l,m,x)=Plm(x)=(−1)m(1−x2)m2dmPl(x)dxm \ No newline at end of file diff --git a/doc/equations/legendre_2.mml b/doc/equations/legendre_2.mml new file mode 100644 index 000000000..16371083d --- /dev/null +++ b/doc/equations/legendre_2.mml @@ -0,0 +1,103 @@ + + + + + + legendre_q + + + 0, + x + + + + = + + + Q + 0 + + + + x + + + + = + + + 1 + 2 + + ln + + + + + 1 + + + x + + + 1 + − + x + + + + + + + + + legendre_q + + + 1, + x + + + + = + + + Q + 1 + + + + x + + + + = + + + x + 2 + + ln + + + + + 1 + + + x + + + 1 + − + x + + + + + − + 1 + + + + + + diff --git a/doc/equations/legendre_2.svg b/doc/equations/legendre_2.svg new file mode 100644 index 000000000..cd5965421 --- /dev/null +++ b/doc/equations/legendre_2.svg @@ -0,0 +1,2 @@ + +legendre_q(0,x)=Q0(x)=12ln(1+x1−x)legendre_q(1,x)=Q1(x)=x2ln(1+x1−x)−1 \ No newline at end of file diff --git a/doc/equations/legendre_3.mml b/doc/equations/legendre_3.mml new file mode 100644 index 000000000..4fffed844 --- /dev/null +++ b/doc/equations/legendre_3.mml @@ -0,0 +1,106 @@ + + + + + + + P + + l + + + − + m + + + + + x + + + + = + + + + + − + 1 + + + m + + + + + + l + − + m + + + ! + + + + + l + + + m + + + ! + + + + P + + l + + m + + + + x + + + + + + + + P + + − + l + − + 1 + + m + + + + x + + + + = + + + P + + l + + m + + + + x + + + + + + + + diff --git a/doc/equations/legendre_3.svg b/doc/equations/legendre_3.svg new file mode 100644 index 000000000..82ccc6c46 --- /dev/null +++ b/doc/equations/legendre_3.svg @@ -0,0 +1,2 @@ + +Pl−m(x)=(−1)m(l−m)!(l+m)!Plm(x)P−l−1m(x)=Plm(x) \ No newline at end of file diff --git a/doc/equations/legendre_4.mml b/doc/equations/legendre_4.mml new file mode 100644 index 000000000..23a5fc294 --- /dev/null +++ b/doc/equations/legendre_4.mml @@ -0,0 +1,65 @@ + + + + P + + l + + + 1 + + + + + x + + + + = + + + + + + 2 + l + + + 1 + + + x + + P + l + + + + x + + + − + l + + P + + l + − + 1 + + + + + x + + + + + + l + + + 1 + + + + + + diff --git a/doc/equations/legendre_4.svg b/doc/equations/legendre_4.svg new file mode 100644 index 000000000..f238ebbd3 --- /dev/null +++ b/doc/equations/legendre_4.svg @@ -0,0 +1,2 @@ + +Pl+1(x)=(2l+1)xPl(x)−lPl−1(x)(l+1) \ No newline at end of file diff --git a/doc/equations/legendre_5.mml b/doc/equations/legendre_5.mml new file mode 100644 index 000000000..a37b8ec41 --- /dev/null +++ b/doc/equations/legendre_5.mml @@ -0,0 +1,80 @@ + + + + P + + l + + + 1 + + m + + + + x + + + + = + + + + + + 2 + l + + + 1 + + + x + + P + + l + + m + + + + x + + + − + + + l + + + m + + + 1 + + + + P + + l + − + 1 + + m + + + + x + + + + + + l + − + m + + + 1 + + + + + + diff --git a/doc/equations/legendre_5.svg b/doc/equations/legendre_5.svg new file mode 100644 index 000000000..d31e4d13d --- /dev/null +++ b/doc/equations/legendre_5.svg @@ -0,0 +1,2 @@ + +Pl+1m(x)=(2l+1)xPlm(x)−(l+m+1)Pl−1m(x)(l−m+1) \ No newline at end of file diff --git a/doc/equations/lgamm1.mml b/doc/equations/lgamm1.mml new file mode 100644 index 000000000..54ccb6766 --- /dev/null +++ b/doc/equations/lgamm1.mml @@ -0,0 +1,21 @@ + + + lgamma + + + z + + + = + ln + | + Γ + + + z + + + | + + + diff --git a/doc/equations/lgamm1.svg b/doc/equations/lgamm1.svg new file mode 100644 index 000000000..bc879a8af --- /dev/null +++ b/doc/equations/lgamm1.svg @@ -0,0 +1,2 @@ + +lgamma(z)=ln|Γ(z)| \ No newline at end of file diff --git a/doc/equations/lgamm2.mml b/doc/equations/lgamm2.mml new file mode 100644 index 000000000..3f3c47cf0 --- /dev/null +++ b/doc/equations/lgamm2.mml @@ -0,0 +1,137 @@ + + + ln + + + | + Γ + + + z + + + | + + + = + z + ln + + + l + + + − + l + + + ln + + + + ∑ + + 0 + + ∞ + + + + + l + n + + + z + + + n + − + + + + + + + + + 1 + + z + − + + + c + 1 + + + + d + 1 + + + + + + c + 2 + + + + d + 2 + + + + + + c + 3 + + + + d + 3 + + + + … + + + + + + + + + + + + ; + + + c + k + + = + − + + + z + + + k + − + 1 + + + + ∧ + + + d + k + + = + z + + + k + + + l + + + diff --git a/doc/equations/lgamm2.svg b/doc/equations/lgamm2.svg new file mode 100644 index 000000000..33e5531b0 --- /dev/null +++ b/doc/equations/lgamm2.svg @@ -0,0 +1,2 @@ + +ln(|Γ(z)|)=zln(l)−l+ln(∑0∞lnzn−+1z−c1d1+c2d2+c3d3+…);ck=−(z+k−1)∧dk=z+k+l \ No newline at end of file diff --git a/doc/equations/lgamm3.mml b/doc/equations/lgamm3.mml new file mode 100644 index 000000000..644b2c39f --- /dev/null +++ b/doc/equations/lgamm3.mml @@ -0,0 +1,52 @@ + + + ln + + + | + Γ + + + − + z + + + | + + + = + ln + + + π + + + − + ln + + + Γ + + + z + + + + + − + ln + + + z + sin + + + π + z + + + + + + + diff --git a/doc/equations/lgamm3.svg b/doc/equations/lgamm3.svg new file mode 100644 index 000000000..eb9cf4322 --- /dev/null +++ b/doc/equations/lgamm3.svg @@ -0,0 +1,2 @@ + +ln(|Γ(−z)|)=ln(π)−ln(Γ(z))−ln(zsin(πz)) \ No newline at end of file diff --git a/doc/equations/lgamm4.mml b/doc/equations/lgamm4.mml new file mode 100644 index 000000000..a3dc8701a --- /dev/null +++ b/doc/equations/lgamm4.mml @@ -0,0 +1,60 @@ + + + ln + + + | + Γ + + + z + + + | + + + = + + + z + − + 0.5 + + + ln + + + + + z + + + g + − + 0.5 + + e + + + + + + ln + + + + L + + e + , + g + + + + + z + + + + + + + diff --git a/doc/equations/lgamm4.svg b/doc/equations/lgamm4.svg new file mode 100644 index 000000000..d15e370ca --- /dev/null +++ b/doc/equations/lgamm4.svg @@ -0,0 +1,2 @@ + +ln(|Γ(z)|)=(z−0.5)ln(z+g−0.5e)+ln(Le,g(z)) \ No newline at end of file diff --git a/doc/equations/lgamm5.mml b/doc/equations/lgamm5.mml new file mode 100644 index 000000000..39dd07ddf --- /dev/null +++ b/doc/equations/lgamm5.mml @@ -0,0 +1,149 @@ + + + ln + + + | + Γ + + + z + + + | + + + = + ? + z + ln + + + + + ? + z + + + g + + + 0.5 + + e + + + + + + + + 1 + 2 + + + ln + + + 1 + + + + + ? + z + + + g + + + 0.5 + + + + + + + ln + + + 1 + + + + ∑ + + k + = + 0 + + + N + − + 1 + + + + + ? + z + + d + k + + + + k + + + ? + z + + + k + + + + + + + + ; + + ? + z + = + 1 + − + z + + ∧ + + + d + k + + = + + + + + g + + + 0.5 + + e + + + + + + + + + C + k + + + + e + g + + + + + + + diff --git a/doc/equations/lgamm5.svg b/doc/equations/lgamm5.svg new file mode 100644 index 000000000..2d4dfc42b --- /dev/null +++ b/doc/equations/lgamm5.svg @@ -0,0 +1,2 @@ + +ln(|Γ(z)|)=?zln(?z+g+0.5e)+12ln(1+?zg+0.5)+ln(1+∑k=0N−1?zdkk(?z+k));?z=1−z∧dk=g+0.5e(Ckeg) \ No newline at end of file diff --git a/doc/equations/lgamm6.mml b/doc/equations/lgamm6.mml new file mode 100644 index 000000000..41e5c603a --- /dev/null +++ b/doc/equations/lgamm6.mml @@ -0,0 +1,158 @@ + + + ln + + + | + Γ + + + z + + + | + + + = + + + 3 + 2 + + + ln + + + 1 + + + + + + ? + z + + + g + + + 1.5 + + + + + + + + ? + z + ln + + + + + z + + + g + − + 0.5 + + e + + + + + + ln + + + 1 + + + + ∑ + + k + = + 1 + + + N + − + 1 + + + + + − + + d + k + + ? + z + + + z + + + k + z + + + + k + 2 + + − + 1 + + + + + + ; + + ? + z + = + z + − + 2 + + ∧ + + + d + k + + = + + + + + + g + + + 1.5 + + e + + + + + 1.5 + + + + + + + + C + k + + + + e + g + + + + + + + diff --git a/doc/equations/lgamm6.svg b/doc/equations/lgamm6.svg new file mode 100644 index 000000000..3c0bf8e34 --- /dev/null +++ b/doc/equations/lgamm6.svg @@ -0,0 +1,2 @@ + +ln(|Γ(z)|)=32ln(1+?zg+1.5)+?zln(z+g−0.5e)+ln(1+∑k=1N−1−dk?zz+kz+k2−1);?z=z−2∧dk=(g+1.5e)1.5(Ckeg) \ No newline at end of file diff --git a/doc/equations/log1pseries.mml b/doc/equations/log1pseries.mml new file mode 100644 index 000000000..688d36d84 --- /dev/null +++ b/doc/equations/log1pseries.mml @@ -0,0 +1,47 @@ + + + log + + + 1 + + + x + + + + = + + + ∑ + + k + = + 1 + + ∞ + + + + + + + − + 1 + + + + k + − + 1 + + + + z + k + + + k + + + + diff --git a/doc/equations/log1pseries.svg b/doc/equations/log1pseries.svg new file mode 100644 index 000000000..0a0d5060e --- /dev/null +++ b/doc/equations/log1pseries.svg @@ -0,0 +1,2 @@ + +log(1+x)=∑k=1∞(−1)k−1zkk \ No newline at end of file diff --git a/doc/equations/lognormal_ref.mml b/doc/equations/lognormal_ref.mml new file mode 100644 index 000000000..845bebc11 --- /dev/null +++ b/doc/equations/lognormal_ref.mml @@ -0,0 +1,55 @@ + + + f + + + x + + + + = + + + 1 + + x + s + + + 2 + π + + + + + + e + + + + − + + + + ln + x + − + m + + + 2 + + + + 2 + + s + 2 + + + + + + + + diff --git a/doc/equations/lognormal_ref.svg b/doc/equations/lognormal_ref.svg new file mode 100644 index 000000000..8e418c9d2 --- /dev/null +++ b/doc/equations/lognormal_ref.svg @@ -0,0 +1,2 @@ + +f(x)=1xs2πe−(lnx−m)22s2 \ No newline at end of file diff --git a/doc/equations/neg_binomial_ref.mml b/doc/equations/neg_binomial_ref.mml new file mode 100644 index 000000000..d29f5e814 --- /dev/null +++ b/doc/equations/neg_binomial_ref.mml @@ -0,0 +1,54 @@ + + + f + + + k + ; + r + , + p + + + + = + + + + Γ + + + r + + + k + + + + + k + ! + Γ + + + r + + + + + + p + r + + + + + 1 + − + p + + + k + + + + diff --git a/doc/equations/neg_binomial_ref.svg b/doc/equations/neg_binomial_ref.svg new file mode 100644 index 000000000..4def4af81 --- /dev/null +++ b/doc/equations/neg_binomial_ref.svg @@ -0,0 +1,2 @@ + +f(k;r,p)=Γ(r+k)k!Γ(r)pr(1−p)k \ No newline at end of file diff --git a/doc/equations/normal_ref1.mml b/doc/equations/normal_ref1.mml new file mode 100644 index 000000000..bc4de02a5 --- /dev/null +++ b/doc/equations/normal_ref1.mml @@ -0,0 +1,57 @@ + + + f + + + x + ; + μ + , + σ + + + + = + + + 1 + + σ + + + 2 + π + + + + + + e + + − + + + + + + x + − + μ + + + 2 + + + + 2 + + σ + 2 + + + + + + + + diff --git a/doc/equations/normal_ref1.svg b/doc/equations/normal_ref1.svg new file mode 100644 index 000000000..1cdf9669d --- /dev/null +++ b/doc/equations/normal_ref1.svg @@ -0,0 +1,2 @@ + +f(x;μ,σ)=1σ2πe−(x−μ)22σ2 \ No newline at end of file diff --git a/doc/equations/poisson_ref1.mml b/doc/equations/poisson_ref1.mml new file mode 100644 index 000000000..3bd7d0910 --- /dev/null +++ b/doc/equations/poisson_ref1.mml @@ -0,0 +1,35 @@ + + + f + + + k + ; + λ + + + + = + + + + + e + + − + λ + + + + λ + k + + + + k + ! + + + + + diff --git a/doc/equations/poisson_ref1.svg b/doc/equations/poisson_ref1.svg new file mode 100644 index 000000000..0f09897ec --- /dev/null +++ b/doc/equations/poisson_ref1.svg @@ -0,0 +1,2 @@ + +f(k;λ)=e−λλkk! \ No newline at end of file diff --git a/doc/equations/roots1.mml b/doc/equations/roots1.mml new file mode 100644 index 000000000..b40b3b32a --- /dev/null +++ b/doc/equations/roots1.mml @@ -0,0 +1,40 @@ + + + + x + + N + + + 1 + + + + = + + + x + N + + − + + + f + + + x + + + + + f + ′ + + + x + + + + + + + diff --git a/doc/equations/roots1.svg b/doc/equations/roots1.svg new file mode 100644 index 000000000..9a5989c6d --- /dev/null +++ b/doc/equations/roots1.svg @@ -0,0 +1,2 @@ + +xN+1=xN−f(x)f′(x) \ No newline at end of file diff --git a/doc/equations/roots2.mml b/doc/equations/roots2.mml new file mode 100644 index 000000000..75244af97 --- /dev/null +++ b/doc/equations/roots2.mml @@ -0,0 +1,70 @@ + + + + x + + N + + + 1 + + + + = + + + x + N + + − + + + 2 + f + + + x + + + f + ′ + + + x + + + + + 2 + + + + f + ′ + + + x + + + + + 2 + + − + f + + + x + + + f + ″ + + + x + + + + + + + diff --git a/doc/equations/roots2.svg b/doc/equations/roots2.svg new file mode 100644 index 000000000..4c7157895 --- /dev/null +++ b/doc/equations/roots2.svg @@ -0,0 +1,2 @@ + +xN+1=xN−2f(x)f′(x)2(f′(x))2−f(x)f″(x) \ No newline at end of file diff --git a/doc/equations/roots3.mml b/doc/equations/roots3.mml new file mode 100644 index 000000000..67dff5e98 --- /dev/null +++ b/doc/equations/roots3.mml @@ -0,0 +1,81 @@ + + + + x + + N + + + 1 + + + + = + + + x + N + + − + + + f + + + x + + + + + f + ′ + + + x + + + + + − + + + f + ″ + + + x + + + + + + f + + + x + + + + + 2 + + + + 2 + + + + f + ′ + + + x + + + + + 3 + + + + + diff --git a/doc/equations/roots3.svg b/doc/equations/roots3.svg new file mode 100644 index 000000000..84eaa5965 --- /dev/null +++ b/doc/equations/roots3.svg @@ -0,0 +1,2 @@ + +xN+1=xN−f(x)f′(x)−f″(x)(f(x))22(f′(x))3 \ No newline at end of file diff --git a/doc/equations/roots4.mml b/doc/equations/roots4.mml new file mode 100644 index 000000000..ae9a4bd0e --- /dev/null +++ b/doc/equations/roots4.mml @@ -0,0 +1,67 @@ + + + + + + f + + + x + + + + + = + + + + x + 3 + + − + a + + + + + f + ′ + + + x + + + + + = + + + 3 + + x + 2 + + + + + + f + ″ + + + x + + + + + = + + + 6 + x + + + + + + diff --git a/doc/equations/roots4.svg b/doc/equations/roots4.svg new file mode 100644 index 000000000..27e8d52a1 --- /dev/null +++ b/doc/equations/roots4.svg @@ -0,0 +1,2 @@ + +f(x)=x3−af′(x)=3x2f″(x)=6x \ No newline at end of file diff --git a/doc/equations/spherical_0.mml b/doc/equations/spherical_0.mml new file mode 100644 index 000000000..72c54cecd --- /dev/null +++ b/doc/equations/spherical_0.mml @@ -0,0 +1,81 @@ + + + + Y + + n + + m + + + + θ + , + φ + + + + = + + + + + + 2 + n + + + 1 + + + 4 + π + + + + + + + n + − + m + + + ! + + + + + n + + + m + + + ! + + + + + + P + + n + + m + + + + cos + θ + + + + e + + ⅈ + m + φ + + + + + diff --git a/doc/equations/spherical_0.svg b/doc/equations/spherical_0.svg new file mode 100644 index 000000000..eb357c794 --- /dev/null +++ b/doc/equations/spherical_0.svg @@ -0,0 +1,2 @@ + +Ynm(θ,φ)=2n+14π(n−m)!(n+m)!Pnm(cosθ)eimφ \ No newline at end of file diff --git a/doc/equations/spherical_1.mml b/doc/equations/spherical_1.mml new file mode 100644 index 000000000..ed28067c2 --- /dev/null +++ b/doc/equations/spherical_1.mml @@ -0,0 +1,85 @@ + + + Re + + + + Y + + n + + m + + + + θ + , + φ + + + + + + = + + + + + + 2 + n + + + 1 + + + 4 + π + + + + + + + n + − + m + + + ! + + + + + n + + + m + + + ! + + + + + + P + + n + + m + + + + cos + θ + + + cos + + + m + φ + + + + + diff --git a/doc/equations/spherical_1.svg b/doc/equations/spherical_1.svg new file mode 100644 index 000000000..bd67f5989 --- /dev/null +++ b/doc/equations/spherical_1.svg @@ -0,0 +1,2 @@ + +Re(Ynm(θ,φ))=2n+14π(n−m)!(n+m)!Pnm(cosθ)cos(mφ) \ No newline at end of file diff --git a/doc/equations/spherical_2.mml b/doc/equations/spherical_2.mml new file mode 100644 index 000000000..4ce0d25e6 --- /dev/null +++ b/doc/equations/spherical_2.mml @@ -0,0 +1,85 @@ + + + Im + + + + Y + + n + + m + + + + θ + , + φ + + + + + + = + + + + + + 2 + n + + + 1 + + + 4 + π + + + + + + + n + − + m + + + ! + + + + + n + + + m + + + ! + + + + + + P + + n + + m + + + + cos + θ + + + sin + + + m + φ + + + + + diff --git a/doc/equations/spherical_2.svg b/doc/equations/spherical_2.svg new file mode 100644 index 000000000..0266a7d5b --- /dev/null +++ b/doc/equations/spherical_2.svg @@ -0,0 +1,2 @@ + +Im(Ynm(θ,φ))=2n+14π(n−m)!(n+m)!Pnm(cosθ)sin(mφ) \ No newline at end of file diff --git a/doc/equations/students_t_dist.mml b/doc/equations/students_t_dist.mml new file mode 100644 index 000000000..c318843e2 --- /dev/null +++ b/doc/equations/students_t_dist.mml @@ -0,0 +1,24 @@ + + + t + + = + + + + μ + − + M + + + + s + + N + + + + + + + diff --git a/doc/equations/students_t_dist.svg b/doc/equations/students_t_dist.svg new file mode 100644 index 000000000..d5d07fecb --- /dev/null +++ b/doc/equations/students_t_dist.svg @@ -0,0 +1,2 @@ + +t=μ−MsN \ No newline at end of file diff --git a/doc/equations/students_t_ref1.mml b/doc/equations/students_t_ref1.mml new file mode 100644 index 000000000..d72b745ab --- /dev/null +++ b/doc/equations/students_t_ref1.mml @@ -0,0 +1,75 @@ + + + f + + + x + + + + = + + + + Γ + + + + + ν + + + 1 + + 2 + + + + + + + + ν + π + + + Γ + + + + ν + 2 + + + + + + + 1 + + + + + + x + 2 + + + ν + + + + + + + ν + + + 1 + + 2 + + + + + + + + diff --git a/doc/equations/students_t_ref1.svg b/doc/equations/students_t_ref1.svg new file mode 100644 index 000000000..b3cd54704 --- /dev/null +++ b/doc/equations/students_t_ref1.svg @@ -0,0 +1,2 @@ + +f(x)=Γ(ν+12)νπΓ(ν2)(1+x2ν)ν+12 \ No newline at end of file