Bessel Functions - 10.15 Derivatives with Respect to Order

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.15.E1	${\displaystyle{\displaystyle\frac{\partial J_{+\nu}\left(z\right)}{\partial\nu}=+J_{+\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{+\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(k+1+\nu\right)}{\Gamma\left(k+1+\nu\right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}}}$ \pderiv{\BesselJ{+\nu}@{z}}{\nu} = +\BesselJ{+\nu}@{z}\ln@{\tfrac{1}{2}z}-(\tfrac{1}{2}z)^{+\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{k+1+\nu}}{\EulerGamma@{k+1+\nu}}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}	${\displaystyle{\displaystyle\Re(k+1+\nu)>0}}$	diff(BesselJ(+ nu, z), nu) = + BesselJ(+ nu, z)*ln((1)/(2)*z)-((1)/(2)*z)^(+ nu)* sum((- 1)^(k)*(Psi(k + 1 + nu))/(GAMMA(k + 1 + nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)	D[BesselJ[+ \[Nu], z], \[Nu]] == + BesselJ[+ \[Nu], z]*Log[Divide[1,2]*z]-(Divide[1,2]*z)^(+ \[Nu])* Sum[(- 1)^(k)*Divide[PolyGamma[k + 1 + \[Nu]],Gamma[k + 1 + \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [7 / 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[2, 3]], Pi]]], Rule[ν, -2]} ... skip entries to safe data
10.15.E1	${\displaystyle{\displaystyle\frac{\partial J_{-\nu}\left(z\right)}{\partial\nu}=-J_{-\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)+(\tfrac{1}{2}z)^{-\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(k+1-\nu\right)}{\Gamma\left(k+1-\nu\right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}}}$ \pderiv{\BesselJ{-\nu}@{z}}{\nu} = -\BesselJ{-\nu}@{z}\ln@{\tfrac{1}{2}z}+(\tfrac{1}{2}z)^{-\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{k+1-\nu}}{\EulerGamma@{k+1-\nu}}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!}	${\displaystyle{\displaystyle\Re(k+1+\nu)>0,\Re((-\nu)+k+1)>0,\Re(k+1-\nu)>0}}$	diff(BesselJ(- nu, z), nu) = - BesselJ(- nu, z)*ln((1)/(2)*z)+((1)/(2)*z)^(- nu)* sum((- 1)^(k)*(Psi(k + 1 - nu))/(GAMMA(k + 1 - nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity)	D[BesselJ[- \[Nu], z], \[Nu]] == - BesselJ[- \[Nu], z]*Log[Divide[1,2]*z]+(Divide[1,2]*z)^(- \[Nu])* Sum[(- 1)^(k)*Divide[PolyGamma[k + 1 - \[Nu]],Gamma[k + 1 - \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [7 / 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[2, 3]], Pi]]], Rule[ν, 2]} ... skip entries to safe data
10.15.E2	${\displaystyle{\displaystyle\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)\left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z\right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{\partial\nu}-\pi J_{\nu}\left(z\right)}}$ \pderiv{\BesselY{\nu}@{z}}{\nu} = \cot@{\nu\pi}\left(\pderiv{\BesselJ{\nu}@{z}}{\nu}-\pi\BesselY{\nu}@{z}\right)-\csc@{\nu\pi}\pderiv{\BesselJ{-\nu}@{z}}{\nu}-\pi\BesselJ{\nu}@{z}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re((-\nu)+k+1)>0}}$	diff(BesselY(nu, z), nu) = cot(nu*Pi)*(diff(BesselJ(nu, z), nu)- Pi*BesselY(nu, z))- csc(nu*Pi)*diff(BesselJ(- nu, z), nu)- Pi*BesselJ(nu, z)	D[BesselY[\[Nu], z], \[Nu]] == Cot[\[Nu]*Pi]*(D[BesselJ[\[Nu], z], \[Nu]]- Pi*BesselY[\[Nu], z])- Csc[\[Nu]*Pi]*D[BesselJ[- \[Nu], z], \[Nu]]- Pi*BesselJ[\[Nu], z]	Successful	Failure	-	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