# numerical differentiation

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## 1—10 of 12 matching pages

##### 2: 3.3 Interpolation
For theory and applications see Stenger (1993, Chapter 3).
##### 3: Philip J. Davis
Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky. …
##### 4: 18.40 Methods of Computation
See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. …
##### 5: Bibliography M
• mpmath (free python library)
• ##### 6: 10.45 Functions of Imaginary Order
In consequence of (10.45.5)–(10.45.7), $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either $\widetilde{I}_{\nu}\left(x\right)$ and $(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)$, or $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu\neq 0$ or $\nu=0$. …
##### 7: 10.24 Functions of Imaginary Order
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). Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either $\widetilde{J}_{\nu}\left(x\right)$ and $\tanh\left(\tfrac{1}{2}\pi\nu\right)\widetilde{Y}_{\nu}\left(x\right)$ or $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$. …
##### 8: 3.7 Ordinary Differential Equations
###### §3.7(ii) Taylor-Series Method: Initial-Value Problems
By repeated differentiation of (3.7.1) all derivatives of $w(z)$ can be expressed in terms of $w(z)$ and $w^{\prime}(z)$ as follows. …
###### §3.7(v) Runge–Kutta Method
The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …
##### 9: 20.14 Methods of Computation
Similarly, their $z$-differentiated forms provide a convenient way of calculating the corresponding derivatives. … In practice a value with, say, $\Im\tau\geq 1/2$, $\left|q\right|\leq 0.2$, is found quickly and is satisfactory for numerical evaluation.
##### 10: 25.11 Hurwitz Zeta Function
25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,$ $s\neq 1$, $\Re s>-1$, $a>0$.
25.11.19 $\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,% \mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x% \right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$.
25.11.20 $(-1)^{k}{\zeta}^{(k)}\left(s,a\right)=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{% 2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k% -r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)% -B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\,\mathrm{d}x+\frac{k(2s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a% \right))^{k-1}}{(x+a)^{s+2}}\,\mathrm{d}x-\frac{k(k-1)}{2}\int_{0}^{\infty}% \frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k-2}}{(x+a% )^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$.
25.11.24 $\sum_{r=1}^{k-1}\zeta'\left(s,\frac{r}{k}\right)=(k^{s}-1)\zeta'\left(s\right)% +k^{s}\zeta\left(s\right)\ln k,$ $s\neq 1$, $k=1,2,3,\dots$.