# with respect to degree or order

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##### 1: 28.12 Definitions and Basic Properties
The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu}\neq 0,1$; equivalently $\nu\neq n$. …
###### §28.12(ii) Eigenfunctions $\mathrm{me}_{\nu}\left(z,q\right)$
For $q=0$, …
##### 3: 14.11 Derivatives with Respect to Degree or Order
###### §14.11 Derivatives with RespecttoDegree or Order
14.11.1 $\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\pi\cot\left(% \nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{1}{\pi}\mathsf{A}_{\nu% }^{\mu}(x),$
14.11.3 $\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)\left(\frac{1+x}{1-x}\right)^% {\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}% \Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{k!\Gamma\left(k-\mu+1% \right)}\*\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right).$
14.11.4 $\left.\frac{\partial}{\partial\mu}\mathsf{P}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{P% }^{\nu}\left(x\right)+\mathsf{Q}^{\nu}\left(x\right),$
14.11.5 $\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=-\tfrac{1}{4}\pi^{2}\mathsf{P}^{\nu}\left(x\right)+\left(\psi\left(-% \nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{Q}^{\nu}\left(x\right).$
##### 4: 1.1 Special Notation
 $x,y$ real variables. … degree. derivatives with respect to the variable, except where indicated otherwise.
##### 5: Bibliography S
• L. Z. Salchev and V. B. Popov (1976) A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.
• B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.
• R. Szmytkowski (2006) On the derivative of the Legendre function of the first kind with respect to its degree. J. Phys. A 39 (49), pp. 15147–15172.
• R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.
• R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.
• ##### 6: 14.20 Conical (or Mehler) Functions
###### §14.20(ix) Asymptotic Approximations: Large $\mu$, $0\leq\tau\leq A\mu$
For the case of purely imaginary order and argument see Dunster (2013).
##### 7: Bibliography C
• C. Chiccoli, S. Lorenzutta, and G. Maino (1990a) An algorithm for exponential integrals of real order. Computing 45 (3), pp. 269–276.
• J. A. Cochran (1965) The zeros of Hankel functions as functions of their order. Numer. Math. 7 (3), pp. 238–250.
• H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
• A. Cruz, J. Esparza, and J. Sesma (1991) Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.
• A. Cruz and J. Sesma (1982) Zeros of the Hankel function of real order and of its derivative. Math. Comp. 39 (160), pp. 639–645.
• ##### 8: 7.23 Tables
• Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$, $x\in[0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\mathrm{d}t$, $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\mathrm{d}t$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

##### 10: 10.24 Functions of Imaginary Order
###### §10.24 Functions of Imaginary Order
and $\widetilde{J}_{\nu}\left(x\right)$, $\widetilde{Y}_{\nu}\left(x\right)$ are linearly independent solutions of (10.24.1): … In consequence of (10.24.6), when $x$ is large $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … …