# partial differentiation

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##### 1: 1.5 Calculus of Two or More Variables
$\frac{{\partial}^{2}f}{\partial y\partial x}=\frac{\partial}{\partial y}\left(% \frac{\partial f}{\partial x}\right).$
1.5.24 $\frac{\mathrm{d}}{\mathrm{d}x}\int^{d}_{c}f(x,y)\mathrm{d}y=\int^{d}_{c}\frac{% \partial f}{\partial x}\mathrm{d}y,$ $a.
##### 3: 36.10 Differential Equations
###### §36.10(ii) Partial Derivatives with Respect to the $x_{n}$
36.10.7 $\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$
36.10.8 $\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{3}}{{\partial y}^{n}},$
36.10.10 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}=i^{n}\frac{{\partial}^{2n}% \Psi_{3}}{{\partial z}^{2n}}.$
##### 4: 25.11 Hurwitz Zeta Function
25.11.17 $\frac{\partial}{\partial a}\zeta\left(s,a\right)=-s\zeta\left(s+1,a\right),$ $s\neq 0,1$; $\Re a>0$.
##### 5: 2.1 Definitions and Elementary Properties
means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left((x-c)^{n}\right)$ as $x\to c$ in $\mathbf{X}$. …
##### 6: 1.9 Calculus of a Complex Variable
###### Differentiation
Differentiability automatically implies continuity.
###### Cauchy–Riemann Equations
Lastly, a power series can be differentiated any number of times within its circle of convergence: …
##### 7: 10.38 Derivatives with Respect to Order
10.38.1 $\frac{\partial I_{\pm\nu}\left(z\right)}{\partial\nu}=\pm I_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}\frac{(% \frac{1}{4}z^{2})^{k}}{k!},$
10.38.2 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}=\tfrac{1}{2}\pi\csc\left(% \nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu}-\frac% {\partial I_{\nu}\left(z\right)}{\partial\nu}\right)-\pi\cot\left(\nu\pi\right% )K_{\nu}\left(z\right),$ $\nu\notin\mathbb{Z}$.
For $\ifrac{\partial I_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4).
10.38.4 $\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\frac{1}{2}z)^{k}K_{k}\left(z% \right)}{k!(n-k)}.$
$\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=0.$
##### 8: 10.15 Derivatives with Respect to Order
10.15.1 $\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
10.15.2 $\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)% \left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z% \right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{% \partial\nu}-\pi J_{\nu}\left(z\right).$
10.15.3 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% \pi}{2}Y_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}% \frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}.$
For $\ifrac{\partial J_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.2.4) and (10.15.3). …
10.15.5 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=\frac{% \pi}{2}Y_{0}\left(z\right),\quad\left.\frac{\partial Y_{\nu}\left(z\right)}{% \partial\nu}\right|_{\nu=0}=-\frac{\pi}{2}J_{0}\left(z\right).$
##### 9: 11.10 Anger–Weber Functions
11.10.27 $\left.\frac{\partial}{\partial\nu}\mathbf{J}_{\nu}\left(z\right)\right|_{\nu=0% }=\tfrac{1}{2}\pi\mathbf{H}_{0}\left(z\right),$
11.10.28 $\left.\frac{\partial}{\partial\nu}\mathbf{E}_{\nu}\left(z\right)\right|_{\nu=0% }=\tfrac{1}{2}\pi J_{0}\left(z\right).$
##### 10: 5.19 Mathematical Applications
By decomposition into partial fractions (§1.2(iii)) …