From 472beb49744eecd3fd61700a112f5a7e4dfe7a08 Mon Sep 17 00:00:00 2001 From: John Maddock Date: Wed, 26 Sep 2007 12:24:35 +0000 Subject: [PATCH] Added mml and svg versions of most of the equations. 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+ + + + + … + + + + + + + + + diff --git a/doc/equations/fraction1.svg b/doc/equations/fraction1.svg new file mode 100644 index 000000000..3de5ddaec --- /dev/null +++ b/doc/equations/fraction1.svg @@ -0,0 +1,2 @@ + +b0+a1b1+a2b2+a3b3+a4b4+… \ No newline at end of file diff --git a/doc/equations/fraction2.mml b/doc/equations/fraction2.mml new file mode 100644 index 000000000..4932943b2 --- /dev/null +++ b/doc/equations/fraction2.mml @@ -0,0 +1,56 @@ + + + + + a + 1 + + + + b + 1 + + + + + + a + 2 + + + + b + 2 + + + + + + a + 3 + + + + b + 3 + + + + + + a + 4 + + + b + 4 + + + + + … + + + + + + + + + diff --git a/doc/equations/fraction2.svg b/doc/equations/fraction2.svg new file mode 100644 index 000000000..14d63bad8 --- /dev/null +++ b/doc/equations/fraction2.svg @@ -0,0 +1,2 @@ + +a1b1+a2b2+a3b3+a4b4+… \ No newline at end of file diff --git a/doc/equations/fraction3.mml b/doc/equations/fraction3.mml new file mode 100644 index 000000000..cf7169439 --- /dev/null +++ b/doc/equations/fraction3.mml @@ -0,0 +1,38 @@ + + + Golden Ratio + + = + + + 1 + + 1 + + + + 1 + + 1 + + + + 1 + + 1 + + + + 1 + + 1 + + + … + + + + + + + + + + + diff --git a/doc/equations/fraction3.svg b/doc/equations/fraction3.svg new file mode 100644 index 000000000..f048f6f31 --- /dev/null +++ b/doc/equations/fraction3.svg @@ -0,0 +1,2 @@ + +Golden Ratio=11+11+11+11+… \ No newline at end of file diff --git a/doc/equations/fraction4.mml b/doc/equations/fraction4.mml new file mode 100644 index 000000000..eaed0f225 --- /dev/null +++ b/doc/equations/fraction4.mml @@ -0,0 +1,62 @@ + + + tan + + + z + + + + = + + + z + + 1 + − + + + z + 2 + + + 3 + − + + + z + 2 + + + 5 + − + + + z + 2 + + + 7 + − + + + z + 2 + + + 9 + − + … + + + + + + + + + + + + + diff --git a/doc/equations/fraction4.svg b/doc/equations/fraction4.svg new file mode 100644 index 000000000..6fe9ed318 --- /dev/null +++ b/doc/equations/fraction4.svg @@ -0,0 +1,2 @@ + +tan(z)=z1−z23−z25−z27−z29−… \ No 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 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+ + + + 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 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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 + 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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 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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 + + + + + + + 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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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@@ -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