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). … ► ►

§3.4(ii) Analytic Functions

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3: 2.1 Definitions and Elementary Properties

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§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 $𝐗$. … ►Differentiation, however, requires the kind of extra conditions needed for the $O$ symbol (§2.1(ii)). …

4: 7.10 Derivatives

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5: 1.5 Calculus of Two or More Variables

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${\displaystyle \frac{{\partial }^{2}f}{\partial y\partial x}}={\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\partial f}{\partial x}}\right).$

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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

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1.5.24 $$\frac{d}{dx}{\int }_c^{d}f(x,y)dy={\int }_c^d\frac{\partial f}{\partial x}dy,$$ .

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(iv), §1.5(iv), §1.5 and Ch.1

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6: 4.10 Integrals

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7: 9.10 Integrals

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9.10.8 $$\int zw(z)dz=w^{\prime }(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.9 $$\int z^{2}w(z)dz=z{w}^{\prime }(z)-w(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.10 $$\int z^{n+3}w(z)dz=z^{n+2}{w}^{\prime }(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)\int z^{n}w(z)dz,$$ $n=0,1,2,\mathrm{\dots }.$

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Defines:

$n$: index (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.20 $${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv=x{\int }_0^x\mathrm{Ai}\left(t\right)dt-{\mathrm{Ai}}^{\prime }\left(x\right)+{\mathrm{Ai}}^{\prime }\left(0\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $x$: real variable and $v$: parameter

Proof sketch:

To verify, differentiate and refer to (9.2.1)

Referenced by:

2nd item, 2nd item

Permalink:

http://dlmf.nist.gov/9.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(viii), §9.10 and Ch.9

►

9.10.21 $${\int }_0^x{\int }_0^v\mathrm{Bi}\left(t\right)dtdv=x{\int }_0^x\mathrm{Bi}\left(t\right)dt-{\mathrm{Bi}}^{\prime }\left(x\right)+{\mathrm{Bi}}^{\prime }\left(0\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $x$: real variable and $v$: parameter

Proof sketch:

To verify, differentiate and refer to (9.2.1)

Referenced by:

2nd item

Permalink:

http://dlmf.nist.gov/9.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(viii), §9.10 and Ch.9

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8: 10.57 Uniform Asymptotic Expansions for Large Order

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9: 20.14 Methods of Computation

… ►Similarly, their $z$-differentiated forms provide a convenient way of calculating the corresponding derivatives. …

10: 23.14 Integrals

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