Bessel Functions - 10.35 Generating Function and Associated Series

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.35.E1	${\displaystyle{\displaystyle e^{\frac{1}{2}z(t+t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}I_{m}\left(z\right)}}$ e^{\frac{1}{2}z(t+t^{-1})} = \sum_{m=-\infty}^{\infty}t^{m}\modBesselI{m}@{z}	${\displaystyle{\displaystyle\Re(m+k+1)>0}}$	exp((1)/(2)*z*(t + (t)^(- 1))) = sum((t)^(m)* BesselI(m, z), m = - infinity..infinity)	Exp[Divide[1,2]*z*(t + (t)^(- 1))] == Sum[(t)^(m)* BesselI[m, z], {m, - Infinity, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
10.35.E2	${\displaystyle{\displaystyle e^{z\cos\theta}=I_{0}\left(z\right)+2\sum_{k=1}^{\infty}I_{k}\left(z\right)\cos\left(k\theta\right)}}$ e^{z\cos@@{\theta}} = \modBesselI{0}@{z}+2\sum_{k=1}^{\infty}\modBesselI{k}@{z}\cos@{k\theta}	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}}$	exp(z*cos(theta)) = BesselI(0, z)+ 2*sum(BesselI(k, z)*cos(k*theta), k = 1..infinity)	Exp[z*Cos[\[Theta]]] == BesselI[0, z]+ 2*Sum[BesselI[k, z]*Cos[k*\[Theta]], {k, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Skipped - Because timed out	Successful [Tested: 70]
10.35.E3	${\displaystyle{\displaystyle e^{z\sin\theta}=I_{0}\left(z\right)+2\sum_{k=0}^{\infty}(-1)^{k}I_{2k+1}\left(z\right)\sin\left((2k+1)\theta\right)+2\sum_{k=1}^{\infty}(-1)^{k}I_{2k}\left(z\right)\cos\left(2k\theta\right)}}$ e^{z\sin@@{\theta}} = \modBesselI{0}@{z}+2\sum_{k=0}^{\infty}(-1)^{k}\modBesselI{2k+1}@{z}\sin@{(2k+1)\theta}+2\sum_{k=1}^{\infty}(-1)^{k}\modBesselI{2k}@{z}\cos@{2k\theta}	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k+1)+k+1)>0,\Re((2k)+k+1)>0}}$	exp(z*sin(theta)) = BesselI(0, z)+ 2*sum((- 1)^(k)* BesselI(2*k + 1, z)*sin((2*k + 1)*theta), k = 0..infinity)+ 2*sum((- 1)^(k)* BesselI(2*k, z)*cos(2*k*theta), k = 1..infinity)	Exp[z*Sin[\[Theta]]] == BesselI[0, z]+ 2*Sum[(- 1)^(k)* BesselI[2*k + 1, z]*Sin[(2*k + 1)*\[Theta]], {k, 0, Infinity}, GenerateConditions->None]+ 2*Sum[(- 1)^(k)* BesselI[2*k, z]*Cos[2*k*\[Theta]], {k, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Skipped - Because timed out
10.35.E4	${\displaystyle{\displaystyle 1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}\left(z\right)-2I_{6}\left(z\right)+\cdots}}$ 1 = \modBesselI{0}@{z}-2\modBesselI{2}@{z}+2\modBesselI{4}@{z}-2\modBesselI{6}@{z}+\dotsb	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(2+k+1)>0,\Re(4+k+1)>0,\Re(6+k+1)>0}}$	1 = BesselI(0, z)- 2*BesselI(2, z)+ 2*BesselI(4, z)- 2*BesselI(6, z)+ ..	1 == BesselI[0, z]- 2*BesselI[2, z]+ 2*BesselI[4, z]- 2*BesselI[6, z]+ \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-9.440290591519046*^-8, -1.7199789187696823*^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-9.924736610669727*^-8, -1.6360842739013975*^-7], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.35.E5	${\displaystyle{\displaystyle e^{+z}=I_{0}\left(z\right)+2I_{1}\left(z\right)+2I_{2}\left(z\right)+2I_{3}\left(z\right)+\cdots}}$ e^{+ z} = \modBesselI{0}@{z}+ 2\modBesselI{1}@{z}+2\modBesselI{2}@{z}+ 2\modBesselI{3}@{z}+\dotsb	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(2+k+1)>0,\Re(3+k+1)>0}}$	exp(+ z) = BesselI(0, z)+ 2*BesselI(1, z)+ 2*BesselI(2, z)+ 2*BesselI(3, z)+ ..	Exp[+ z] == BesselI[0, z]+ 2*BesselI[1, z]+ 2*BesselI[2, z]+ 2*BesselI[3, z]+ \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-0.003384051289485407, 0.00475177611436145], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.002576303532707505, 0.004074841322498801], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.35.E5	${\displaystyle{\displaystyle e^{-z}=I_{0}\left(z\right)-2I_{1}\left(z\right)+2I_{2}\left(z\right)-2I_{3}\left(z\right)+\cdots}}$ e^{- z} = \modBesselI{0}@{z}- 2\modBesselI{1}@{z}+2\modBesselI{2}@{z}- 2\modBesselI{3}@{z}+\dotsb	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(2+k+1)>0,\Re(3+k+1)>0}}$	exp(- z) = BesselI(0, z)- 2*BesselI(1, z)+ 2*BesselI(2, z)- 2*BesselI(3, z)+ ..	Exp[- z] == BesselI[0, z]- 2*BesselI[1, z]+ 2*BesselI[2, z]- 2*BesselI[3, z]+ \[Ellipsis]	Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-0.0024389937896763803, 0.0042567403420422645], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.0020316532349716754, 0.004934003265463338], Times[-1.0, …]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data