Is there any known formulae for the derivative of the Bessel function with respect to the order of the Bessel function?
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series... from Maple $${\frac {d}{dr}}{{\rm J}_r\left(2\right)}= \sum _{k=0}^{\infty }{\frac { \left( -1 \right) ^{k+1}\Psi \left( 1+r+k \right) }{\Gamma \left( 1+r+k \right) \Gamma \left( 1+k \right) }}$$ |
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Abramowitz and Stegun give a couple of special cases but don't give a general result. Starting from some of the integral or series representations and differentiating you can get a corresponding integral or series for the derivative, but I would guess that it's unlikely to simplify to a "known" function in the general case. An example they give is (for the spherical Bessel function [ d/d\nu j_\nu(x) ]_{\nu=0} = (\pi/(2x))(Ci(2x)\sin x - SI(2x)\cos x) They also give examples evaluated at |
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I suggest looking at Landau's paper:http://www.emis.de/journals/EJDE/conf-proc/04/l1/landau.pdf |
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As I understand it you are looking for $D_\nu(x):=\frac{d}{d\nu} J_\nu(x)$. Perhaps you would like to explain a bit why you are looking at $D_\nu$? I have worked a bit with the Legendre functions of the first ($P_\nu$) and second kind ($Q_\nu$). Where my primary interest was to find estimates in $\nu$ and both parameters. To find such estimates basically I used relations together with integral representations. At one point I estimated an integral expression for $\frac{d}{d\nu} Q_\nu(x)$ in order to see that for fixed $x>1$ it is decreasing with respect to $\nu$. (The main reason to these studies was to prove a Tauberian theorem for spaces like $L^1_w(G//K)$ - the double coset space of $G=SL(2,R)$ - where $\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx$ is the Fourier transform.) |
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