partial differentiation

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1: 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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1.5.24 $$\frac{d}{dx}{\int }_c^{d}f(x,y)dy={\int }_c^d\frac{\partial f}{\partial x}dy,$$ .

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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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2: 3.4 Differentiation

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§3.4(iii) Partial Derivatives

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3: 36.10 Differential Equations

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§36.10(ii) Partial Derivatives with Respect to the $x_n$

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36.10.7 $$\frac{{\partial }^{2n}{\mathrm{\Psi }}_2}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_2}{{\partial y}^n}.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

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36.10.8 $$\frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial x}^{2n}}={\mathrm{i}}^n\frac{{\partial }^n{\mathrm{\Psi }}_3}{{\partial y}^n},$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $m$: integer and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

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36.10.10 $$\frac{{\partial }^{3n}{\mathrm{\Psi }}_3}{{\partial y}^{3n}}={\mathrm{i}}^n\frac{{\partial }^{2n}{\mathrm{\Psi }}_3}{{\partial z}^{2n}}.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/36.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(ii), §36.10(ii), §36.10 and Ch.36

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§36.10(iv) Partial $z$-Derivatives

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4: 25.11 Hurwitz Zeta Function

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25.11.17 $$\frac{\partial }{\partial a}\zeta (s,a)=-s\zeta (s+1,a),$$ $s\ne 0,1$; $\mathrm{\Re }a>0$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $a$: real or complex parameter and $s$: complex variable

Keywords:

derivative

Proof sketch:

Derivable by differentiating (25.11.1) with respect to $a$.

Referenced by:

(25.11.10)

Permalink:

http://dlmf.nist.gov/25.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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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 $𝐗$. …

6: 1.9 Calculus of a Complex Variable

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§1.9(ii) Continuity, Point Sets, and Differentiation

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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

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10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). ►

10.38.4 $${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$

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Symbols:

$!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.45

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=-K_0\left(z\right),$

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10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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8: 10.15 Derivatives with Respect to Order

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10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.2 $$\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }=\cot \left(\nu \pi \right)\left(\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }-\pi Y_{\nu }\left(z\right)\right)-\csc \left(\nu \pi \right)\frac{\partial J_{-\nu }\left(z\right)}{\partial \nu }-\pi J_{\nu }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.65

Referenced by:

(10.15.1), §10.15

Permalink:

http://dlmf.nist.gov/10.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►For $\partial J_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.2.4) and (10.15.3). … ►

10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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9: 11.10 Anger–Weber Functions

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11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.28 $${\frac{\partial }{\partial \nu }{𝐄}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi J_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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10: 5.19 Mathematical Applications

… ►By decomposition into partial fractions (§1.2(iii)) …