Weierstrass Sigma Function

Author

Eric W. Weisstein
December 12, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/WeierstrassSigmaFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/WeierstrassSigmaFunction.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Notation

WeierstrassSigma[z,{(Subscript[g, 2]),(Subscript[g, 3])}]//TraditionalForm

Plot

<<Utilities`Plot`

ComplexPlot3D[WeierstrassSigma[z,{4,0}],{z,-4,4},PlotPoints->100]//Timing

ComplexContourPlot[WeierstrassSigma[z,{4,0}],{z,-4,4}]//Timing

Derivative

D[Log@WeierstrassSigma[z,{g2,g3}],z]

D[WeierstrassSigma[z,{g2,g3}],z]

Series

Series[WeierstrassSigma[z,{g2,g3}],{z,0,2}]

Terms

a[0,0]:=1
a[m_Integer?Negative,n_]:=0
a[n_,m_Integer?Negative]:=0

a[m_,n_]:=a[m,n]=3(m+1)a[m+1,n-1]+16/3(n+1)a[m-2,n+1]-1/3(2m+3n-1)(4m+6n-1)a[m-1,n]

a[0,0]

Table[Table[a[m,n],{n,0,5}],{m,0,5}]

c[2]:=(Subscript[c, 2])
c[3]:=(Subscript[c, 3])
c[k_]:=c[k]=3/(2k+1)/(k-3)Sum[c[m]c[k-m],{m,2,k-2}]

Table[FullSimplify[c[n]],{n,4,10}]//TableForm//TraditionalForm

Limit

should be 1

Limit[WeierstrassSigma[z,{g2,g3}]/z,z->0]