Entire Function

Author

Eric W. Weisstein
January 25, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/EntireFunction.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Tests

Reduce`HolomorphicFunctionQ[#,x,{-Infinity,Infinity}]&/@{WeberE[x,x^2-1],DawsonF[x],AngerJ[x/5,Sin[x]]}

Reduce`HolomorphicFunctionQ[Cos[x],x,{-Infinity,Infinity}]

Reduce`HolomorphicFunctionQ[EllipticTheta[2,z+1,z],z,{-10(I+1),10(I+1)}]

Reduce`HolomorphicFunctionQ[EllipticTheta[2,z+1,z],z,{-Infinity(I+1),Infinity(I+1)}]

Grid[{#[z],Reduce`HolomorphicFunctionQ[#[z],n,{-Infinity(I+1),Infinity(I+1)}]}&/@{AiryAi,AiryBi,AngerJ[n,#]&,BarnesG,KelvinBei,KelvinBer,BesselJ[n,#]&,BesselY[n,#]&,Cos,Cosh,Coversine,Erf,Erfc,Erfi,Exp,FresnelC,FresnelS,(*1/Gamma[#]&,*)
EllipticTheta[1,n,#]&,EllipticTheta[2,z+1,#]&,EllipticTheta[3,n,#]&,EllipticTheta[4,n,#]&,EllipticThetaPrime[1,n,#]&,EllipticThetaPrime[2,n,#]&,EllipticThetaPrime[3,n,#]&,EllipticThetaPrime[4,n,#]&,Haversine,Hyperfactorial,LogBarnesG,SinhIntegral,Sin,SinIntegral,Sinh,StruveL[n,#]&,BesselJ[n,#]&,StruveH[n,#]&,Versine,WeberE[n,#]&},
Alignment->Left]//TraditionalForm

Reduce`HolomorphicFunctionQ[Sin[x],x,{-Infinity(I+1),Infinity(I+1)}]

Reduce`HolomorphicFunctionQ[Tan[x],x,{-Infinity(I+1),Infinity(I+1)}]