The complex components:

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\,\mathrm{argument}\right)$

$\left[\arg \left(z\right)=\mathrm{-I}\ln \left(\frac{z}{\left|z\right|}\right)\&comma;\arg \left(z\right)=\mathrm{-I}\ln \left(\mathrm{signum}\left(z\right)\right)\&comma;\arg \left(z\right)=\arctan \left(\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)\right)\&comma;\left[\arg \left(za\right)=\arg \left(z\right)\&comma;0<a\right]\&comma;\arg \left(za\right)=\arg \left(z\right)+\arg \left(a\right)+2\mathrm{\pi }⌊\frac{1}{2}-\frac{\arg \left(z\right)}{2\mathrm{\pi }}-\frac{\arg \left(a\right)}{2\mathrm{\pi }}⌋\&comma;\arg \left(\frac{z}{a}\right)=\arg \left(z\right)-\arg \left(a\right)+2\mathrm{\pi }⌊\frac{1}{2}-\frac{\arg \left(z\right)}{2\mathrm{\pi }}+\frac{\arg \left(a\right)}{2\mathrm{\pi }}⌋\&comma;\left[\arg \left(z^a\right)=a\arg \left(z\right)\&comma;a::\mathrm{real}\mathbf{And}-\mathrm{\pi }<a\arg \left(z\right)\mathbf{And}a\arg \left(z\right)<\mathrm{\pi }\right]\&comma;\left[\arg \left(z^a\right)=\arg \left({\&ExponentialE;}^{\mathrm{I}a\arg \left(z\right)}\right)\&comma;a::\mathrm{real}\right]\&comma;\arg \left(z^a\right)=\arctan \left(\sin \left(\arctan \left(\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)\right)\mathrm{\Re }\left(a\right)+\mathrm{\Im }\left(a\right)\ln \left(\left|z\right|\right)\right)\&comma;\cos \left(\arctan \left(\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)\right)\mathrm{\Re }\left(a\right)+\mathrm{\Im }\left(a\right)\ln \left(\left|z\right|\right)\right)\right)\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\&comma;\mathrm{Re}\right)$

$\left[\mathrm{\Re }\left(\mathrm{I}z\right)=-\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(z\right)=\frac{z\left(1+\frac{1}{{\&ExponentialE;}^{2\mathrm{I}\arg \left(z\right)}}\right)}{2}\&comma;\mathrm{\Re }\left(z\right)=\frac{z\left(1+\frac{1}{{\mathrm{signum}\left(z\right)}^2}\right)}{2}\&comma;\mathrm{\Re }\left(z\right)=\frac{z}{2}+\frac{{\left|z\right|}^2}{2z}\&comma;\mathrm{\Re }\left(z\right)=\frac{z}{2}+\frac{\stackrel{\&conjugate0;}{z}}{2}\&comma;\left[\mathrm{\Re }\left(za\right)=a\mathrm{\Re }\left(z\right)\&comma;a::\mathrm{real}\right]\&comma;\mathrm{\Re }\left(za\right)=\mathrm{\Re }\left(z\right)\mathrm{\Re }\left(a\right)-\mathrm{\Im }\left(a\right)\mathrm{\Im }\left(z\right)\&comma;\mathrm{\Re }\left(\frac{z}{a}\right)=\frac{\mathrm{\Re }\left(z\right)\mathrm{\Re }\left(a\right)+\mathrm{\Im }\left(a\right)\mathrm{\Im }\left(z\right)}{{\left|a\right|}^2}\&comma;\mathrm{\Re }\left(z^a\right)={\left|z\right|}^{\mathrm{\Re }\left(a\right)}{\&ExponentialE;}^{-\arg \left(z\right)\mathrm{\Im }\left(a\right)}\cos \left(\arg \left(z\right)\mathrm{\Re }\left(a\right)+\mathrm{\Im }\left(a\right)\ln \left(\left|z\right|\right)\right)\right]$

The Jacobi elliptic functions:

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\&comma;\mathrm{JacobiAM}\right)$

$\left[\left[\mathrm{am}\left(z|k\right)=\arccos \left(\mathrm{cn}\left(z|k\right)\right)\&comma;0<\mathrm{\Re }\left(\mathrm{am}\left(z|k\right)\right)<\mathrm{\pi }\mathbf{And}k\in \left[\mathrm{-1}\&comma;1\right]\mathbf{or}\mathrm{\Re }\left(\mathrm{am}\left(z|k\right)\right)=0\mathbf{And}0\le \mathrm{\Im }\left(\mathrm{am}\left(z|k\right)\right)\mathbf{And}k\in \left[\mathrm{-1}\&comma;1\right]\mathbf{or}\mathrm{\Re }\left(\mathrm{am}\left(z|k\right)\right)=\mathrm{\pi }\mathbf{And}\mathrm{\Im }\left(\mathrm{am}\left(z|k\right)\right)\le 0\mathbf{And}k\in \left[\mathrm{-1}\&comma;1\right]\right]\&comma;\left[\mathrm{am}\left(z|k\right)=\arcsin \left(\mathrm{sn}\left(z|k\right)\right)\&comma;-\frac{\mathrm{\pi }}{2}<\mathrm{\Re }\left(\mathrm{am}\left(z|k\right)\right)<\frac{\mathrm{\pi }}{2}\mathbf{And}k::\mathrm{real}\mathbf{or}\mathrm{\Re }\left(\mathrm{am}\left(z|k\right)\right)=-\frac{\mathrm{\pi }}{2}\mathbf{And}0\le \mathrm{\Im }\left(\mathrm{am}\left(z|k\right)\right)\mathbf{And}k::\mathrm{real}\mathbf{or}\mathrm{\Re }\left(\mathrm{am}\left(z|k\right)\right)=\frac{\mathrm{\pi }}{2}\mathbf{And}\mathrm{\Im }\left(\mathrm{am}\left(z|k\right)\right)\le 0\mathbf{And}k::\mathrm{real}\right]\right]$

A new topic, periodicity, got added:

$\mathrm{FunctionAdvisor}\left(\mathrm{periodicity}\&comma;\mathrm{JacobiAM}\right)$

$\left[\left[\mathrm{am}\left(2\mathit{K}\left(k\right)m+z|k\right)=\mathrm{am}\left(z|k\right)+\mathrm{\pi }\left\{\begin{array}{cc}m& \left|k\right|\le 1\\ \left\{\begin{array}{cc}0& m::\mathrm{even}\\ \mathrm{signum}\left(m\right)& \mathrm{otherwise}\end{array}\right.& \mathrm{otherwise}\end{array}\right.\&comma;\mathrm{And}\left(m::\mathrm{integer}\right)\right]\&comma;\left[\mathrm{am}\left(2\mathrm{I}\mathit{K}\prime \left(k\right)n+z|k\right)={\left(\mathrm{-1}\right)}^n\mathrm{am}\left(z|k\right)+\mathrm{\pi }\left\{\begin{array}{cc}0& n::\mathrm{even}\\ \mathrm{signum}\left(\mathrm{\Re }\left(z\right)\right)\mathrm{signum}\left(n\right)& \mathrm{otherwise}\end{array}\right.\&comma;\mathrm{And}\left(n::\mathrm{integer}\right)\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{periodicity}\&comma;\mathrm{JacobiCN}\right)$

$\left[\mathrm{cn}\left(z+2\mathit{K}\left(k\right)r+2\mathrm{I}\mathit{K}\prime \left(k\right)s|k\right)={\left(\mathrm{-1}\right)}^{s+r}\mathrm{cn}\left(z|k\right)\&comma;r::\mathrm{integer}\mathbf{And}s::\mathrm{integer}\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\&comma;\mathrm{JacobiCN}\right)$

$\left[\mathrm{cn}\left(z|k\right)=\frac{\mathrm{\pi }\left(\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(\mathrm{-1}\right)}^n\mathrm{sech}\left(\frac{\mathrm{\pi }\mathit{K}\left(k\right)\left(n+\frac{z}{2\mathit{K}\left(k\right)}\right)}{\mathit{K}\prime \left(k\right)}\right)\right)}{2\sqrt{k^2}\mathit{K}\prime \left(k\right)}\&comma;\mathrm{cn}\left(z+\mathit{K}\left(k\right)|k\right)=-\frac{\sqrt{-k^2+1}\mathrm{sn}\left(z|k\right)}{\mathrm{dn}\left(z|k\right)}\&comma;\mathrm{cn}\left(z+\mathrm{I}\mathit{K}\prime \left(k\right)|k\right)=\frac{\mathrm{-I}\mathrm{dn}\left(z|k\right)}{\sqrt{k^2}\mathrm{sn}\left(z|k\right)}\&comma;\mathrm{cn}\left(z|\frac{1}{k}\right)=\mathrm{dn}\left(\frac{z}{k}|k\right)\&comma;\mathrm{cn}\left(\mathrm{I}z|k\right)=\frac{1}{\mathrm{cn}\left(z|\sqrt{-k^2+1}\right)}\&comma;\mathrm{cn}\left(z|\mathrm{I}k\right)=\mathrm{dc}\left(kz|\sqrt{1+\frac{1}{k^2}}\right)\&comma;\mathrm{cn}\left(\left(1+k\right)z|2\sqrt{\frac{k}{{\left(1+k\right)}^2}}\right)=\frac{\mathrm{cn}\left(z|k\right)\mathrm{dn}\left(z|k\right)}{1+k{\mathrm{sn}\left(z|k\right)}^2}\&comma;\mathrm{cn}\left(z|k\right)=\mathrm{nc}\left(\mathrm{I}z|\sqrt{-k^2+1}\right)\&comma;\mathrm{cn}\left(\left(1+\sqrt{-k^2+1}\right)z|\sqrt{\frac{{\left(1-\sqrt{-k^2+1}\right)}^2}{{\left(1+\sqrt{-k^2+1}\right)}^2}}\right)=\frac{1-\left(1+\sqrt{-k^2+1}\right){\mathrm{sn}\left(z|k\right)}^2}{\mathrm{dn}\left(z|k\right)}\&comma;{\mathrm{cn}\left(\frac{z}{2}|k\right)}^2=\frac{\mathrm{dn}\left(z|k\right)+\mathrm{cn}\left(z|k\right)}{1+\mathrm{dn}\left(z|k\right)}\&comma;{\mathrm{cn}\left(z|k\right)}^2=1-{\mathrm{sn}\left(z|k\right)}^2\&comma;\mathrm{cn}\left(v+z|k\right)=\frac{\mathrm{sn}\left(z|k\right)\mathrm{dn}\left(z|k\right)\mathrm{sn}\left(v|k\right)\mathrm{dn}\left(v|k\right)-\mathrm{cn}\left(z|k\right)\mathrm{cn}\left(v|k\right)}{k^2{\mathrm{sn}\left(v|k\right)}^2{\mathrm{sn}\left(z|k\right)}^2-1}\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\&comma;\mathrm{InverseJacobiSN}\right)$

The JacobiTheta functions:

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\&comma;\mathrm{JacobiTheta1}\right)$

$\left[\left[{\mathrm{\vartheta }}_1\left(z+\frac{\left(2m+1\right)\mathrm{\pi }}{2}\&comma;q\right)={\left(\mathrm{-1}\right)}^m{\mathrm{\vartheta }}_2\left(z\&comma;q\right)\&comma;m::\mathrm{integer}\right]\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;q\left(k\right)\right)={\left(k^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{sn}\left(\frac{2\mathit{K}\left(k\right)z}{\mathrm{\pi }}|k\right){\mathrm{\vartheta }}_4\left(z\&comma;q\left(k\right)\right)\&comma;\left[{\mathrm{\vartheta }}_1\left(\frac{z}{\mathrm{\tau }}\&comma;{\&ExponentialE;}^{\frac{\mathrm{-I}\mathrm{\pi }}{\mathrm{\tau }}}\right)=\mathrm{-I}\sqrt{\mathrm{-I}\mathrm{\tau }}{\mathrm{\vartheta }}_1\left(z\&comma;{\&ExponentialE;}^{\mathrm{I}\mathrm{\pi }\mathrm{\tau }}\right){\left({\&ExponentialE;}^{\frac{\mathrm{-I}\mathrm{\pi }}{\mathrm{\tau }}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\&ExponentialE;}^{\frac{\mathrm{I}{z}^2}{\mathrm{\pi }\mathrm{\tau }}+\frac{\mathrm{I}\mathrm{\pi }}{4\mathrm{\tau }}}\&comma;0<\mathrm{\Im }\left(\mathrm{\tau }\right)\right]\&comma;{{\mathrm{\vartheta }}_1\left(z\&comma;q\right)}^2=\frac{{{\mathrm{\vartheta }}_2\left(0\&comma;q\right)}^2{{\mathrm{\vartheta }}_3\left(z\&comma;q\right)}^2-{{\mathrm{\vartheta }}_3\left(0\&comma;q\right)}^2{{\mathrm{\vartheta }}_2\left(z\&comma;q\right)}^2}{{{\mathrm{\vartheta }}_4\left(0\&comma;q\right)}^2}\&comma;{{\mathrm{\vartheta }}_2\left(0\&comma;q\right)}^2{\mathrm{\vartheta }}_1\left(y+x\&comma;q\right){\mathrm{\vartheta }}_1\left(x-y\&comma;q\right)={{\mathrm{\vartheta }}_1\left(x\&comma;q\right)}^2{{\mathrm{\vartheta }}_2\left(y\&comma;q\right)}^2-{{\mathrm{\vartheta }}_2\left(x\&comma;q\right)}^2{{\mathrm{\vartheta }}_1\left(y\&comma;q\right)}^2\&comma;{\mathrm{\vartheta }}_1\left(2z\&comma;q\right)=\frac{2{\mathrm{\vartheta }}_1\left(z\&comma;q\right){\mathrm{\vartheta }}_2\left(z\&comma;q\right){\mathrm{\vartheta }}_3\left(z\&comma;q\right){\mathrm{\vartheta }}_4\left(z\&comma;q\right)}{{\mathrm{\vartheta }}_2\left(0\&comma;q\right){\mathrm{\vartheta }}_3\left(0\&comma;q\right){\mathrm{\vartheta }}_4\left(0\&comma;q\right)}\&comma;{\mathrm{\vartheta }}_1\left(y+x\&comma;q\right){\mathrm{\vartheta }}_1\left(x-y\&comma;q\right){\mathrm{\vartheta }}_1\left(c+d\&comma;q\right){\mathrm{\vartheta }}_1\left(c-d\&comma;q\right)-{\mathrm{\vartheta }}_1\left(x+c\&comma;q\right){\mathrm{\vartheta }}_1\left(-x+c\&comma;q\right){\mathrm{\vartheta }}_1\left(d+y\&comma;q\right){\mathrm{\vartheta }}_1\left(d-y\&comma;q\right)+{\mathrm{\vartheta }}_1\left(x+d\&comma;q\right){\mathrm{\vartheta }}_1\left(-x+d\&comma;q\right){\mathrm{\vartheta }}_1\left(y+c\&comma;q\right){\mathrm{\vartheta }}_1\left(-y+c\&comma;q\right)=0\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;\sqrt{q}\right)=\frac{2{\mathrm{\vartheta }}_1\left(z\&comma;q\right){\mathrm{\vartheta }}_4\left(z\&comma;q\right)}{{\mathrm{\vartheta }}_2\left(0\&comma;\sqrt{q}\right)}\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;q\right)=2q^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sin \left(z\right)\left(\prod _{k=1}^{\mathrm{\infty }}\left(1-q^{2k}\right)\left(1-2q^{2k}\cos \left(2z\right)+q^{4k}\right)\right)\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;q\right)={\mathrm{\vartheta }}_2\left(z-\frac{\mathrm{\pi }}{2}\&comma;q\right)\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;q\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_3\left(z+\frac{\mathrm{I}\ln \left(q\right)}{2}-\frac{\mathrm{\pi }}{2}\&comma;q\right)q^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}z}}\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;q\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_4\left(z+\frac{\mathrm{I}\ln \left(q\right)}{2}\&comma;q\right)q^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}z}}\right]$

The Weierstrass functions:

$\mathrm{FunctionAdvisor}\left(\mathrm{identities}\&comma;\mathrm{WeierstrassP}\right)$

$\left[𝒫\left(\mathrm{I}z;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=-𝒫\left(z;\mathrm{g\_\_2},-\mathrm{g\_\_3}\right)\&comma;𝒫\left(z;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=a^2𝒫\left(za;\frac{\mathrm{g\_\_2}}{a^4},\frac{\mathrm{g\_\_3}}{a^6}\right)\&comma;𝒫\left(\mathrm{z\_\_1}+\mathrm{z\_\_2};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=\frac{{\left(𝒫\prime \left(\mathrm{z\_\_1};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)-𝒫\prime \left(\mathrm{z\_\_2};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)\right)}^2}{4{\left(𝒫\left(\mathrm{z\_\_1};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)-𝒫\left(\mathrm{z\_\_2};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)\right)}^2}-𝒫\left(\mathrm{z\_\_1};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)-𝒫\left(\mathrm{z\_\_2};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)\&comma;\left[𝒫\left(\mathrm{\&omega;\_\_j};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=\mathrm{e\_\_j}\&comma;j\in \left[1\&comma;2\&comma;3\right]\mathbf{And}{z}^3-\frac{1}{4}\mathrm{g\_\_2}z-\frac{1}{4}\mathrm{g\_\_3}=\left(z-\mathrm{e\_\_1}\right)\left(z-\mathrm{e\_\_2}\right)\left(z-\mathrm{e\_\_3}\right)\mathbf{And}\mathrm{\&omega;\_\_1}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_2}-\mathrm{e\_\_3}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\mathbf{And}\mathrm{\&omega;\_\_3}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_1}-\mathrm{e\_\_2}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_3}-\mathrm{e\_\_1}}}\mathbf{And}\mathrm{\&omega;\_\_2}=-\mathrm{\&omega;\_\_3}-\mathrm{\&omega;\_\_1}\mathbf{And}\left(\frac{{\mathrm{g\_\_2}}^3}{{\mathrm{g\_\_3}}^2}\ne 27\mathbf{Or}\left(\mathrm{e\_\_1}=t\mathbf{And}\mathrm{g\_\_2}=3t^2\mathbf{And}\mathrm{g\_\_3}=t^3\right)\right)\mathbf{And}\left(\mathrm{g\_\_2}\ne 0\mathbf{Or}\mathrm{g\_\_3}\ne 0\right)\right]\&comma;\left[𝒫\left(z;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=\mathrm{e\_\_1}+\frac{{\mathrm{\pi }}^2{{\mathrm{\vartheta }}_3\left(0\&comma;{\&ExponentialE;}^{\frac{\mathrm{I}\mathrm{\pi }\mathrm{\&omega;\_\_3}}{\mathrm{\&omega;\_\_1}}}\right)}^2{{\mathrm{\vartheta }}_4\left(0\&comma;{\&ExponentialE;}^{\frac{\mathrm{I}\mathrm{\pi }\mathrm{\&omega;\_\_3}}{\mathrm{\&omega;\_\_1}}}\right)}^2{{\mathrm{\vartheta }}_2\left(\frac{\mathrm{\pi }z}{2\mathrm{\&omega;\_\_1}}\&comma;{\&ExponentialE;}^{\frac{\mathrm{I}\mathrm{\pi }\mathrm{\&omega;\_\_3}}{\mathrm{\&omega;\_\_1}}}\right)}^2}{4{\mathrm{\&omega;\_\_1}}^2{{\mathrm{\vartheta }}_1\left(\frac{\mathrm{\pi }z}{2\mathrm{\&omega;\_\_1}}\&comma;{\&ExponentialE;}^{\frac{\mathrm{I}\mathrm{\pi }\mathrm{\&omega;\_\_3}}{\mathrm{\&omega;\_\_1}}}\right)}^2}\&comma;z^3-\frac{1}{4}\mathrm{g\_\_2}z-\frac{1}{4}\mathrm{g\_\_3}=\left(z-\mathrm{e\_\_1}\right)\left(z-\mathrm{e\_\_2}\right)\left(z-\mathrm{e\_\_3}\right)\mathbf{And}\mathrm{\&omega;\_\_1}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_2}-\mathrm{e\_\_3}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\mathbf{And}\mathrm{\&omega;\_\_3}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_1}-\mathrm{e\_\_2}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_3}-\mathrm{e\_\_1}}}\mathbf{And}\left(\frac{{\mathrm{g\_\_2}}^3}{{\mathrm{g\_\_3}}^2}\ne 27\mathbf{Or}\left(\mathrm{e\_\_1}=t\mathbf{And}\mathrm{g\_\_2}=3t^2\mathbf{And}\mathrm{g\_\_3}=t^3\right)\right)\mathbf{And}\left(\mathrm{g\_\_2}\ne 0\mathbf{Or}\mathrm{g\_\_3}\ne 0\right)\right]\&comma;\left[𝒫\left(v;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)𝒫\prime \left(w;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)-𝒫\prime \left(v;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)𝒫\left(w;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)-𝒫\left(u;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)𝒫\prime \left(w;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)+𝒫\prime \left(u;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)𝒫\left(w;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)+𝒫\left(u;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)𝒫\prime \left(v;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)-𝒫\prime \left(u;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)𝒫\left(v;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=0\&comma;\mathrm{And}\left(u+v+w=0\right)\right]\&comma;\left[n^2𝒫\left(nz;\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=\sum _{j=0}^{n-1}\sum _{l=0}^{n-1}𝒫\left(z+\frac{2j\mathrm{\&omega;\_\_1}}{n}+\frac{2l\mathrm{\&omega;\_\_3}}{n};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)\&comma;z^3-\frac{1}{4}\mathrm{g\_\_2}z-\frac{1}{4}\mathrm{g\_\_3}=\left(z-\mathrm{e\_\_1}\right)\left(z-\mathrm{e\_\_2}\right)\left(z-\mathrm{e\_\_3}\right)\mathbf{And}\mathrm{\&omega;\_\_1}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_2}-\mathrm{e\_\_3}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\mathbf{And}\mathrm{\&omega;\_\_3}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_1}-\mathrm{e\_\_2}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_3}-\mathrm{e\_\_1}}}\mathbf{And}\left(\frac{{\mathrm{g\_\_2}}^3}{{\mathrm{g\_\_3}}^2}\ne 27\mathbf{Or}\left(\mathrm{e\_\_1}=t\mathbf{And}\mathrm{g\_\_2}=3t^2\mathbf{And}\mathrm{g\_\_3}=t^3\right)\right)\mathbf{And}\left(\mathrm{g\_\_2}\ne 0\mathbf{Or}\mathrm{g\_\_3}\ne 0\right)\right]\&comma;\left[𝒫\left(\frac{\mathrm{\&omega;\_\_1}}{2};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=\mathrm{e\_\_1}+\left(\mathrm{e\_\_1}-\mathrm{e\_\_3}\right)\mathrm{csgn}\left(\sqrt{\frac{\mathrm{e\_\_1}-\mathrm{e\_\_2}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)\sqrt{\frac{\mathrm{e\_\_1}-\mathrm{e\_\_2}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\&comma;z^3-\frac{1}{4}\mathrm{g\_\_2}z-\frac{1}{4}\mathrm{g\_\_3}=\left(z-\mathrm{e\_\_1}\right)\left(z-\mathrm{e\_\_2}\right)\left(z-\mathrm{e\_\_3}\right)\mathbf{And}\mathrm{\&omega;\_\_1}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_2}-\mathrm{e\_\_3}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\mathbf{And}\left(\frac{{\mathrm{g\_\_2}}^3}{{\mathrm{g\_\_3}}^2}\ne 27\mathbf{Or}\left(\mathrm{e\_\_1}=t\mathbf{And}\mathrm{g\_\_2}=3t^2\mathbf{And}\mathrm{g\_\_3}=t^3\right)\right)\mathbf{And}\left(\mathrm{g\_\_2}\ne 0\mathbf{Or}\mathrm{g\_\_3}\ne 0\right)\right]\&comma;\left[𝒫\left(\frac{\mathrm{\&omega;\_\_3}}{2};\mathrm{g\_\_2},\mathrm{g\_\_3}\right)=\mathrm{e\_\_3}-\left(\mathrm{e\_\_1}-\mathrm{e\_\_3}\right)\mathrm{csgn}\left(\sqrt{\frac{\mathrm{e\_\_2}-\mathrm{e\_\_3}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)\sqrt{\frac{\mathrm{e\_\_2}-\mathrm{e\_\_3}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\&comma;z^3-\frac{1}{4}\mathrm{g\_\_2}z-\frac{1}{4}\mathrm{g\_\_3}=\left(z-\mathrm{e\_\_1}\right)\left(z-\mathrm{e\_\_2}\right)\left(z-\mathrm{e\_\_3}\right)\mathbf{And}\mathrm{\&omega;\_\_3}=\sqrt{\frac{{\mathit{K}\left(\sqrt{\frac{\mathrm{e\_\_1}-\mathrm{e\_\_2}}{\mathrm{e\_\_1}-\mathrm{e\_\_3}}}\right)}^2}{\mathrm{e\_\_3}-\mathrm{e\_\_1}}}\mathbf{And}\left(\frac{{\mathrm{g\_\_2}}^3}{{\mathrm{g\_\_3}}^2}\ne 27\mathbf{Or}\left(\mathrm{e\_\_1}=t\mathbf{And}\mathrm{g\_\_2}=3t^2\mathbf{And}\mathrm{g\_\_3}=t^3\right)\right)\mathbf{And}\left(\mathrm{g\_\_2}\ne 0\mathbf{Or}\mathrm{g\_\_3}\ne 0\right)\right]\right]$

New specializations for elliptic functions:

$\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\&comma; \mathrm{JacobiSN}\&comma; \mathrm{JacobiCN}\right)$

$\left[\mathrm{sn}\left(z|k\right)=\sqrt{1-{\mathrm{cn}\left(z|k\right)}^2}\&comma;\mathrm{And}\left(0<\mathrm{\Re }\left(\mathrm{sn}\left(z|k\right)\right)\right)\mathbf{or}\mathrm{\Re }\left(\mathrm{sn}\left(z|k\right)\right)=0\mathbf{And}0<\mathrm{\Im }\left(\mathrm{sn}\left(z|k\right)\right)\right],\left[\mathrm{sn}\left(z|k\right)=-\sqrt{1-{\mathrm{cn}\left(z|k\right)}^2}\&comma;\mathrm{And}\left(\mathrm{\Re }\left(\mathrm{sn}\left(z|k\right)\right)<0\right)\mathbf{or}\mathrm{\Re }\left(\mathrm{sn}\left(z|k\right)\right)=0\mathbf{And}\mathrm{\Im }\left(\mathrm{sn}\left(z|k\right)\right)<0\right]$

New conversion routines in connection with these enhancements to the FunctionAdvisor.

$\mathrm{FunctionAdvisor}\left(\mathrm{jacobi}\right)$

* Partial match of "jacobi" against topic "Jacobi_related".
The 18 functions in the "Jacobi_related" class are:

$\left[\mathrm{JacobiAM}\&comma;\mathrm{JacobiCD}\&comma;\mathrm{JacobiCN}\&comma;\mathrm{JacobiCS}\&comma;\mathrm{JacobiDC}\&comma;\mathrm{JacobiDN}\&comma;\mathrm{JacobiDS}\&comma;\mathrm{JacobiNC}\&comma;\mathrm{JacobiND}\&comma;\mathrm{JacobiNS}\&comma;\mathrm{JacobiSC}\&comma;\mathrm{JacobiSD}\&comma;\mathrm{JacobiSN}\&comma;\mathrm{JacobiTheta1}\&comma;\mathrm{JacobiTheta2}\&comma;\mathrm{JacobiTheta3}\&comma;\mathrm{JacobiTheta4}\&comma;\mathrm{JacobiZeta}\right]$

(2.1.1)(z,k)$$

$\left[\mathrm{am}\left(z|k\right)\&comma;\mathrm{cd}\left(z|k\right)\&comma;\mathrm{cn}\left(z|k\right)\&comma;\mathrm{cs}\left(z|k\right)\&comma;\mathrm{dc}\left(z|k\right)\&comma;\mathrm{dn}\left(z|k\right)\&comma;\mathrm{ds}\left(z|k\right)\&comma;\mathrm{nc}\left(z|k\right)\&comma;\mathrm{nd}\left(z|k\right)\&comma;\mathrm{ns}\left(z|k\right)\&comma;\mathrm{sc}\left(z|k\right)\&comma;\mathrm{sd}\left(z|k\right)\&comma;\mathrm{sn}\left(z|k\right)\&comma;{\mathrm{\vartheta }}_1\left(z\&comma;k\right)\&comma;{\mathrm{\vartheta }}_2\left(z\&comma;k\right)\&comma;{\mathrm{\vartheta }}_3\left(z\&comma;k\right)\&comma;{\mathrm{\vartheta }}_4\left(z\&comma;k\right)\&comma;Z\left(z\&comma;k\right)\right]$

$\left[14\right]$

$\mathrm{JacobiTheta1}$

$$\begin{gathered} \mathbf{for} m \mathbf{from} 14 \mathbf{to} 17 \mathbf{do} \mathbf{for} n \mathbf{from}\mathbf{ }\mathbf{14}\mathbf{ }\mathbf{to} 17 \mathbf{do} \\ \mathbf{if} m \ne n \mathbf{then} \mathrm{print}\left(\left[m\right] \&equals; \mathrm{convert}\left(\left[m\right]\&comma;\left[n\right]\right)\right) \mathbf{fi} \\ \mathbf{od}\&semi; \mathbf{od}\&semi; \end{gathered}$$

${\mathrm{\vartheta }}_1\left(z\&comma;k\right)={\mathrm{\vartheta }}_2\left(z-\frac{\mathrm{\pi }}{2}\&comma;k\right)$

${\mathrm{\vartheta }}_1\left(z\&comma;k\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_3\left(z+\frac{\mathrm{I}\ln \left(k\right)}{2}-\frac{\mathrm{\pi }}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}z}}$

${\mathrm{\vartheta }}_1\left(z\&comma;k\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_4\left(z+\frac{\mathrm{I}\ln \left(k\right)}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}z}}$

${\mathrm{\vartheta }}_2\left(z\&comma;k\right)={\mathrm{\vartheta }}_1\left(z+\frac{\mathrm{\pi }}{2}\&comma;k\right)$

${\mathrm{\vartheta }}_2\left(z\&comma;k\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_3\left(z+\frac{\mathrm{I}\ln \left(k\right)}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}\left(z+\frac{\mathrm{\pi }}{2}\right)}}$

${\mathrm{\vartheta }}_2\left(z\&comma;k\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_4\left(z+\frac{\mathrm{\pi }}{2}+\frac{\mathrm{I}\ln \left(k\right)}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}\left(z+\frac{\mathrm{\pi }}{2}\right)}}$

${\mathrm{\vartheta }}_3\left(z\&comma;k\right)={\mathrm{\vartheta }}_1\left(z-\frac{\mathrm{I}\ln \left(k\right)}{2}+\frac{\mathrm{\pi }}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\&ExponentialE;}^{\mathrm{I}z}$

${\mathrm{\vartheta }}_3\left(z\&comma;k\right)={\mathrm{\vartheta }}_2\left(z-\frac{\mathrm{I}\ln \left(k\right)}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\&ExponentialE;}^{\mathrm{I}z}$

${\mathrm{\vartheta }}_3\left(z\&comma;k\right)=\mathrm{I}{\mathrm{\vartheta }}_4\left(z+\frac{\mathrm{\pi }}{2}\&comma;k\right)\sqrt{k}{\&ExponentialE;}^{-\frac{\ln \left(k\right)}{2}-\frac{\mathrm{I}\mathrm{\pi }}{2}}$

${\mathrm{\vartheta }}_4\left(z\&comma;k\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_1\left(z+\frac{\mathrm{I}\ln \left(k\right)}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}z}}$

${\mathrm{\vartheta }}_4\left(z\&comma;k\right)=\frac{\mathrm{I}{\mathrm{\vartheta }}_2\left(z+\frac{\mathrm{I}\ln \left(k\right)}{2}-\frac{\mathrm{\pi }}{2}\&comma;k\right)k^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{{\&ExponentialE;}^{\mathrm{I}z}}$

${\mathrm{\vartheta }}_4\left(z\&comma;k\right)=-\frac{{\mathrm{\vartheta }}_3\left(z+\mathrm{I}\ln \left(k\right)-\frac{\mathrm{\pi }}{2}\&comma;k\right)\sqrt{k}}{{\&ExponentialE;}^{-\frac{\ln \left(k\right)}{2}+2\mathrm{I}z}}$

$\mathbf{for} m \mathbf{from} 2 \mathbf{to} 13 \mathbf{do} \left[m\right] \&equals;\mathrm{convert}\left(\left[m\right]\&comma; \mathrm{JacobiSN}\right) \mathbf{od}\&semi;$

$\mathrm{cd}\left(z|k\right)=\frac{\mathrm{I}\mathrm{sn}\left(z\sqrt{-k^2+1}|\frac{\mathrm{I}k}{\sqrt{-k^2+1}}\right)}{\mathrm{sn}\left(\mathrm{I}z\sqrt{-k^2+1}|\frac{1}{\sqrt{-k^2+1}}\right)}$

$\mathrm{cn}\left(z|k\right)=\frac{\mathrm{I}\mathrm{sn}\left(z|k\right)}{\mathrm{sn}\left(\mathrm{I}z|\sqrt{-k^2+1}\right)}$

$\mathrm{cs}\left(z|k\right)=\frac{\mathrm{I}}{\mathrm{sn}\left(\mathrm{I}z|\sqrt{-k^2+1}\right)}$

$\mathrm{dc}\left(z|k\right)=\frac{\mathrm{-I}\mathrm{sn}\left(\mathrm{I}z\sqrt{-k^2+1}|\frac{1}{\sqrt{-k^2+1}}\right)}{\mathrm{sn}\left(z\sqrt{-k^2+1}|\frac{\mathrm{I}k}{\sqrt{-k^2+1}}\right)}$

$\mathrm{dn}\left(z|k\right)=\frac{\mathrm{I}\mathrm{sn}\left(kz|\frac{1}{k}\right)}{\mathrm{sn}\left(\mathrm{I}kz|\frac{\mathrm{I}\sqrt{-k^2+1}}{k}\right)}$

$\mathrm{ds}\left(z|k\right)=\frac{\mathrm{I}\mathrm{sn}\left(kz|\frac{1}{k}\right)}{\mathrm{sn}\left(\mathrm{I}kz|\frac{\mathrm{I}\sqrt{-k^2+1}}{k}\right)\mathrm{sn}\left(z|k\right)}$

$\mathrm{nc}\left(z|k\right)=\frac{\mathrm{-I}\mathrm{sn}\left(\mathrm{I}z|\sqrt{-k^2+1}\right)}{\mathrm{sn}\left(z|k\right)}$

$\mathrm{nd}\left(z|k\right)=\frac{\mathrm{-I}\mathrm{sn}\left(\mathrm{I}kz|\frac{\mathrm{I}\sqrt{-k^2+1}}{k}\right)}{\mathrm{sn}\left(kz|\frac{1}{k}\right)}$

$\mathrm{ns}\left(z|k\right)=\frac{1}{\mathrm{sn}\left(z|k\right)}$

$\mathrm{sc}\left(z|k\right)=\mathrm{-I}\mathrm{sn}\left(\mathrm{I}z|\sqrt{-k^2+1}\right)$

$\mathrm{sd}\left(z|k\right)=\frac{\mathrm{-I}\mathrm{sn}\left(\mathrm{I}kz|\frac{\mathrm{I}\sqrt{-k^2+1}}{k}\right)\mathrm{sn}\left(z|k\right)}{\mathrm{sn}\left(kz|\frac{1}{k}\right)}$

$\mathrm{sn}\left(z|k\right)=\mathrm{sn}\left(z|k\right)$

The MathematicalFunctions package has 7 new commands.

$\mathrm{with}\left(\mathrm{MathematicalFunctions}\right)$

$\left[\mathrm{\&Intersect}\&comma;\mathrm{\&Minus}\&comma;\mathrm{\&Union}\&comma;\mathrm{Assume}\&comma;\mathrm{Coulditbe}\&comma;\mathrm{Evalf}\&comma;\mathrm{Get}\&comma;\mathrm{Is}\&comma;\mathrm{SearchFunction}\&comma;\mathrm{Series}\right]$

$\mathrm{R\_\_1} ≔x \mathbf{in} \mathrm{RealRange}\left(-2\&comma;\mathrm{infinity}\right)$

$\mathrm{R\_\_1}≔x\in \left[\mathrm{-2}\&comma;\mathrm{\infty }\right]$

$\mathrm{R\_\_2}≔x \mathbf{in} \mathrm{RealRange}\left(-\mathrm{infinity}\&comma; 5\right)$

$\mathrm{R\_\_2}≔x\in \left[-\mathrm{\infty }\&comma;5\right]$

$\mathrm{R\_\_1}\&Minus\mathrm{R\_\_2}$

$5<x\le \mathrm{\infty }$

$\mathrm{R\_\_1}\&Intersect\mathrm{R\_\_2}$

$\mathrm{-2}\le x\le 5$

$\mathrm{R\_\_3} ≔ 0<x^2+\left(3x\right)+2$

$\mathrm{R\_\_3}≔0<x^2+3x+2$

$\mathrm{R\_\_4} ≔ x\&Hat;2-9 \ge 0$

$\mathrm{R\_\_4}≔0\le x^2-9$

$\mathrm{R\_\_3} \&amp;\mathrm{Union} \mathrm{R\_\_4}$

$x<\mathrm{-2}\mathbf{Or}\mathrm{-1}<x$

$\mathrm{R\_\_3} \&amp;\mathrm{Intersect} \mathrm{R\_\_4}$

$x\le \mathrm{-3}\mathbf{Or}3\le x$

$\mathrm{R\_\_3} \&amp;\mathrm{Minus} \mathrm{R\_\_4}$

$\left(\mathrm{-3}<x<\mathrm{-2}\right)\mathbf{Or}\left(\mathrm{-1}<x<3\right)$

Assume uses the new Physics:-Assume command, capable of placing assumptions without redefining the variables, hence it is possible to reuse expressions entered before placing the assumptions. The result can also be reused after removing the assumptions, computing as in an extended-assuming context.

$\mathrm{simplify}\left(\arccos \left(\cos \left(x\right)\right)\right)$

$\arccos \left(\cos \left(x\right)\right)$

$\mathrm{Assume}\left(0 < x < \frac{\mathrm{\pi }}{2}\right)$

$\left\{x::\left(0\&comma;\frac{\mathrm{\pi }}{2}\right)\right\}$

$\mathrm{simplify}\left(\right)$

$x$

Coulditbe and Is permit computing the related operations with an arbitrary number of conditions to be tested, that could be nested with no restrictions, involving And and Or, and also lists.

$\mathrm{c\_\_1} ≔ \mathrm{And}\left(b < a\&comma; b \&equals; a\right)$

$\mathrm{c\_\_1}≔b<a\mathbf{And}b=a$

$\mathrm{c\_\_2} ≔ \mathrm{And}\left(b \&gt; a\&comma; b \&equals; a\right)$

$\mathrm{c\_\_2}≔a<b\mathbf{And}b=a$

$\mathrm{Coulditbe}\left(\mathrm{Or}\left(\mathrm{c\_\_1}\&comma;\mathrm{c\_\_2}\right)\right)$