# differentiation

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##### 1: 27.20 Methods of Computation: Other Number-Theoretic Functions
A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau\left(n\right)$, and the values can be checked by the congruence (27.14.20). …
##### 2: 3.4 Differentiation
###### §3.4 Differentiation
and follows from the differentiated form of (3.3.4). …
##### 3: 2.1 Definitions and Elementary Properties
###### §2.1(ii) Integration and Differentiation
Differentiation requires extra conditions. …This result also holds with both $O$’s replaced by $o$’s. … 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 $\mathbf{X}$. … Differentiation, however, requires the kind of extra conditions needed for the $O$ symbol (§2.1(ii)). …
##### 5: 1.8 Fourier Series
###### §1.8(iii) Integration and Differentiation
If a function $f(x)\in C^{2}[0,2\pi]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime}(x)$. …
##### 7: 9.10 Integrals
9.10.8 $\int zw(z)\mathrm{d}z=w^{\prime}(z),$
9.10.9 $\int z^{2}w(z)\mathrm{d}z=zw^{\prime}(z)-w(z),$
9.10.10 $\int z^{n+3}w(z)\mathrm{d}z=z^{n+2}w^{\prime}(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)% \int z^{n}w(z)\mathrm{d}z,$ $n=0,1,2,\ldots.$
9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\mathrm{Ai}\left(t\right)\mathrm{d}t\mathrm{d}v=x% \int_{0}^{x}\mathrm{Ai}\left(t\right)\mathrm{d}t-\mathrm{Ai}'\left(x\right)+% \mathrm{Ai}'\left(0\right),$
9.10.21 $\int_{0}^{x}\!\!\int_{0}^{v}\mathrm{Bi}\left(t\right)\mathrm{d}t\mathrm{d}v=x% \int_{0}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t-\mathrm{Bi}'\left(x\right)+% \mathrm{Bi}'\left(0\right),$
##### 9: 20.14 Methods of Computation
Similarly, their $z$-differentiated forms provide a convenient way of calculating the corresponding derivatives. …