Bessel Functions - 10.27 Connection Formulas

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.27.E1	${\displaystyle{\displaystyle I_{-n}\left(z\right)=I_{n}\left(z\right)}}$ \modBesselI{-n}@{z} = \modBesselI{n}@{z}	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	BesselI(- n, z) = BesselI(n, z)	BesselI[- n, z] == BesselI[n, z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.27.E2	${\displaystyle{\displaystyle I_{-\nu}\left(z\right)=I_{\nu}\left(z\right)+(2/\pi)\sin\left(\nu\pi\right)K_{\nu}\left(z\right)}}$ \modBesselI{-\nu}@{z} = \modBesselI{\nu}@{z}+(2/\pi)\sin@{\nu\pi}\modBesselK{\nu}@{z}	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}}$	BesselI(- nu, z) = BesselI(nu, z)+(2/Pi)*sin(nu*Pi)*BesselK(nu, z)	BesselI[- \[Nu], z] == BesselI[\[Nu], z]+(2/Pi)*Sin[\[Nu]*Pi]*BesselK[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
10.27.E3	${\displaystyle{\displaystyle K_{-\nu}\left(z\right)=K_{\nu}\left(z\right)}}$ \modBesselK{-\nu}@{z} = \modBesselK{\nu}@{z}		BesselK(- nu, z) = BesselK(nu, z)	BesselK[- \[Nu], z] == BesselK[\[Nu], z]	Successful	Successful	-	Successful [Tested: 70]
10.27.E4	${\displaystyle{\displaystyle K_{\nu}\left(z\right)=\tfrac{1}{2}\pi\frac{I_{-\nu}\left(z\right)-I_{\nu}\left(z\right)}{\sin\left(\nu\pi\right)}}}$ \modBesselK{\nu}@{z} = \tfrac{1}{2}\pi\frac{\modBesselI{-\nu}@{z}-\modBesselI{\nu}@{z}}{\sin@{\nu\pi}}	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}}$	BesselK(nu, z) = (1)/(2)*Pi*(BesselI(- nu, z)- BesselI(nu, z))/(sin(nu*Pi))	BesselK[\[Nu], z] == Divide[1,2]*Pi*Divide[BesselI[- \[Nu], z]- BesselI[\[Nu], z],Sin[\[Nu]*Pi]]	Successful	Successful	-	Failed [14 / 70] Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.27.E6	${\displaystyle{\displaystyle I_{\nu}\left(z\right)=e^{-\nu\pi i/2}J_{\nu}\left(ze^{+\pi i/2}\right)}}$ \modBesselI{\nu}@{z} = e^{-\nu\pi i/2}\BesselJ{\nu}@{ze^{+\pi i/2}}	${\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{2}\pi,\Re(\nu+k+1)>0}}$	BesselI(nu, z) = exp(- nu*Pi*I/2)*BesselJ(nu, z*exp(+ Pi*I/2))	BesselI[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselJ[\[Nu], z*Exp[+ Pi*I/2]]	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E6	${\displaystyle{\displaystyle I_{\nu}\left(z\right)=e^{+\nu\pi i/2}J_{\nu}\left(ze^{-\pi i/2}\right)}}$ \modBesselI{\nu}@{z} = e^{+\nu\pi i/2}\BesselJ{\nu}@{ze^{-\pi i/2}}	${\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{2}\pi,\Re(\nu+k+1)>0}}$	BesselI(nu, z) = exp(+ nu*Pi*I/2)*BesselJ(nu, z*exp(- Pi*I/2))	BesselI[\[Nu], z] == Exp[+ \[Nu]*Pi*I/2]*BesselJ[\[Nu], z*Exp[- Pi*I/2]]	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E7	${\displaystyle{\displaystyle I_{\nu}\left(z\right)=\tfrac{1}{2}e^{-\nu\pi i/2}\left({H^{(1)}_{\nu}}\left(ze^{+\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{+\pi i/2}\right)\right)}}$ \modBesselI{\nu}@{z} = \tfrac{1}{2}e^{-\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{+\pi i/2}}+\HankelH{2}{\nu}@{ze^{+\pi i/2}}\right)	${\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{2}\pi,\Re(\nu+k+1)>0}}$	BesselI(nu, z) = (1)/(2)*exp(- nu*Pi*I/2)*(HankelH1(nu, z*exp(+ Pi*I/2))+ HankelH2(nu, z*exp(+ Pi*I/2)))	BesselI[\[Nu], z] == Divide[1,2]*Exp[- \[Nu]*Pi*I/2]*(HankelH1[\[Nu], z*Exp[+ Pi*I/2]]+ HankelH2[\[Nu], z*Exp[+ Pi*I/2]])	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E7	${\displaystyle{\displaystyle I_{\nu}\left(z\right)=\tfrac{1}{2}e^{+\nu\pi i/2}\left({H^{(1)}_{\nu}}\left(ze^{-\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{-\pi i/2}\right)\right)}}$ \modBesselI{\nu}@{z} = \tfrac{1}{2}e^{+\nu\pi i/2}\left(\HankelH{1}{\nu}@{ze^{-\pi i/2}}+\HankelH{2}{\nu}@{ze^{-\pi i/2}}\right)	${\displaystyle{\displaystyle-\pi\leq+\operatorname{ph}z,-\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\tfrac{1}{2}\pi,-\operatorname{ph}z\leq\tfrac{1}{2}\pi,\Re(\nu+k+1)>0}}$	BesselI(nu, z) = (1)/(2)*exp(+ nu*Pi*I/2)*(HankelH1(nu, z*exp(- Pi*I/2))+ HankelH2(nu, z*exp(- Pi*I/2)))	BesselI[\[Nu], z] == Divide[1,2]*Exp[+ \[Nu]*Pi*I/2]*(HankelH1[\[Nu], z*Exp[- Pi*I/2]]+ HankelH2[\[Nu], z*Exp[- Pi*I/2]])	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E9	${\displaystyle{\displaystyle\pi iJ_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)-e^{\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right)}}$ \pi i\BesselJ{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}-e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi,\Re(\nu+k+1)>0}}$	Pi*I*BesselJ(nu, z) = exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))- exp(nu*Pi*I/2)*BesselK(nu, z*exp(Pi*I/2))	Pi*I*BesselJ[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]- Exp[\[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[Pi*I/2]]	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E10	${\displaystyle{\displaystyle-\pi Y_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)+e^{\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right)}}$ -\pi\BesselY{\nu}@{z} = e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}+e^{\nu\pi i/2}\modBesselK{\nu}@{ze^{\pi i/2}}	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{2}\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	- Pi*BesselY(nu, z) = exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))+ exp(nu*Pi*I/2)*BesselK(nu, z*exp(Pi*I/2))	- Pi*BesselY[\[Nu], z] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]+ Exp[\[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[Pi*I/2]]	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E11	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=e^{+(\nu+1)\pi i/2}I_{\nu}\left(ze^{-\pi i/2}\right)-(2/\pi)e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)}}$ \BesselY{\nu}@{z} = e^{+(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{-\pi i/2}}-(2/\pi)e^{-\nu\pi i/2}\modBesselK{\nu}@{ze^{-\pi i/2}}	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi\leq+\operatorname{ph}z,-\tfrac{1}{2}\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\pi,-\operatorname{ph}z\leq\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = exp(+(nu + 1)*Pi*I/2)*BesselI(nu, z*exp(- Pi*I/2))-(2/Pi)*exp(- nu*Pi*I/2)*BesselK(nu, z*exp(- Pi*I/2))	BesselY[\[Nu], z] == Exp[+(\[Nu]+ 1)*Pi*I/2]*BesselI[\[Nu], z*Exp[- Pi*I/2]]-(2/Pi)*Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[- Pi*I/2]]	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]
10.27.E11	${\displaystyle{\displaystyle Y_{\nu}\left(z\right)=e^{-(\nu+1)\pi i/2}I_{\nu}\left(ze^{+\pi i/2}\right)-(2/\pi)e^{+\nu\pi i/2}K_{\nu}\left(ze^{+\pi i/2}\right)}}$ \BesselY{\nu}@{z} = e^{-(\nu+1)\pi i/2}\modBesselI{\nu}@{ze^{+\pi i/2}}-(2/\pi)e^{+\nu\pi i/2}\modBesselK{\nu}@{ze^{+\pi i/2}}	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi\leq+\operatorname{ph}z,-\tfrac{1}{2}\pi\leq-\operatorname{ph}z,+\operatorname{ph}z\leq\pi,-\operatorname{ph}z\leq\pi,\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	BesselY(nu, z) = exp(-(nu + 1)*Pi*I/2)*BesselI(nu, z*exp(+ Pi*I/2))-(2/Pi)*exp(+ nu*Pi*I/2)*BesselK(nu, z*exp(+ Pi*I/2))	BesselY[\[Nu], z] == Exp[-(\[Nu]+ 1)*Pi*I/2]*BesselI[\[Nu], z*Exp[+ Pi*I/2]]-(2/Pi)*Exp[+ \[Nu]*Pi*I/2]*BesselK[\[Nu], z*Exp[+ Pi*I/2]]	Failure	Failure	Successful [Tested: 50]	Successful [Tested: 50]