holomorphic differentials

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11: 1.6 Vectors and Vector-Valued Functions

… ►

1.6.19 $$\nabla =𝐢\frac{\partial }{\partial x}+𝐣\frac{\partial }{\partial y}+𝐤\frac{\partial }{\partial z}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: variable, $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Permalink:

http://dlmf.nist.gov/1.6.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(iii), §1.6(iii), §1.6 and Ch.1

… ►where $d𝐒$ is the surface element with an attached normal direction ${𝐓}_u\times {𝐓}_v$. … ►Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. … ►

1.6.57 $${\iint }_S(\nabla \times 𝐅)\cdot d𝐒={\int }_{\partial S}𝐅\cdot d𝐬,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\partial x$: partial differential of $x$ and $S$: parameterized surface

Permalink:

http://dlmf.nist.gov/1.6.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

… ►where $\partial g/\partial n=\nabla g\cdot 𝐧$ is the derivative of $g$ normal to the surface outwards from $V$ and $𝐧$ is the unit outer normal vector. …

12: 4.40 Integrals

… ►

4.40.1 $$\int \sinh xdx=\cosh x,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.77

Permalink:

http://dlmf.nist.gov/4.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

►

4.40.2 $$\int \cosh xdx=\sinh x,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.78

Permalink:

http://dlmf.nist.gov/4.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

►

4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

… ►

4.40.5 $$\int \mathrm{sech}xdx=\mathrm{gd}\left(x\right).$$

ⓘ

Symbols:

$\mathrm{gd}x$: Gudermannian function, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.81

Permalink:

http://dlmf.nist.gov/4.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

… ►

4.40.6 $$\int \coth xdx=\ln \left(\sinh x\right),$$ .

ⓘ

Symbols:

$dx$: differential of $x$ , $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.82

Permalink:

http://dlmf.nist.gov/4.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

…

13: 6.14 Integrals

… ►

6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

►

6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

►

6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

… ►

6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $E_1\left(z\right)$: exponential integral, $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

5.1.33

Permalink:

http://dlmf.nist.gov/6.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

… ►

6.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{Ci}}^2\left(t\right)dt={\int }_0^{\mathrm{\infty }}{\mathrm{si}}^2\left(t\right)dt=\frac{1}{2}\pi ,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.31

Permalink:

http://dlmf.nist.gov/6.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

…

14: 4.26 Integrals

… ►

4.26.1 $$\int \sin xdx=-\cos x,$$

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.113

Permalink:

http://dlmf.nist.gov/4.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

►

4.26.2 $$\int \cos xdx=\sin x.$$

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.114

Permalink:

http://dlmf.nist.gov/4.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

… ►

4.26.9 $${\int }_0^{\pi }\sin \left(mt\right)\sin \left(nt\right)dt=0,$$ $m\ne n$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $m$: integer and $n$: integer

A&S Ref:

4.3.140

Permalink:

http://dlmf.nist.gov/4.26.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

►

4.26.10 $${\int }_0^{\pi }\cos \left(mt\right)\cos \left(nt\right)dt=0,$$ $m\ne n$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $m$: integer and $n$: integer

A&S Ref:

4.3.140

Permalink:

http://dlmf.nist.gov/4.26.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

… ►

4.26.14 $$\int \arcsin xdx=x\arcsin x+{(1-x^2)}^{1/2},$$ ,

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.4.58 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

…

15: 10.15 Derivatives with Respect to Order

… ►

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

ⓘ

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

… ►

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

ⓘ

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). ►

10.15.4 $${\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-\frac{\pi }{2}{J}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{Y}_k\left(z\right)}{k!(n-k)},$$

ⓘ

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:

9.1.67

Permalink:

http://dlmf.nist.gov/10.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

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

ⓘ

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

…

16: 12.17 Physical Applications

… ►

12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►

12.17.4 $$\frac{1}{{\xi }^2+{\eta }^2}\left(\frac{{\partial }^{2}w}{{\partial \xi }^2}+\frac{{\partial }^{2}w}{{\partial \eta }^2}\right)+\frac{{\partial }^{2}w}{{\partial \zeta }^2}+k^{2}w=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: constant, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/12.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. …

17: 9.11 Products

… ►

§9.11(i) Differential Equation

… ►

9.11.5 $$\int w_1{w}_{2}dz=-w_1^{\prime }{w}_2^{\prime }+z{w}_1{w}_2,$$

ⓘ

Symbols:

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

Source:

Albright (1977, (A.16))

Permalink:

http://dlmf.nist.gov/9.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

►

9.11.6 $$\int w_1{w}_2^{\prime }dz=\frac{1}{2}\left(w_1{w}_2+z𝒲\left\{w_1,w_2\right\}\right),$$

ⓘ

Symbols:

$𝒲$: Wronskian, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: function

Source:

Albright (1977, (A.17))

Permalink:

http://dlmf.nist.gov/9.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

►

9.11.7 $$\int w_1^{\prime }{w}_2^{\prime }dz=\frac{1}{3}(w_1{w}_2^{\prime }+w_1^{\prime }{w}_2+z{w}_1^{\prime }{w}_2^{\prime }-z^2{w}_1{w}_2),$$

ⓘ

Symbols:

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

Source:

Albright (1977, (A.18))

Permalink:

http://dlmf.nist.gov/9.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

… ►For $\int z^n{w}_1{w}_{2}dz$, $\int z^n{w}_1{w}_2^{\prime }dz$, $\int z^n{w}_1^{\prime }{w}_2^{\prime }dz$, where $n$ is any positive integer, see Albright (1977). …

18: 10.73 Physical Applications

… ►

10.73.1 $${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{{\partial \varphi }^2}+\frac{{\partial }^{2}V}{{\partial z}^2}=0,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.2 $${\nabla }^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{{\partial t}^2},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation ►

10.73.3 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►

10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(ii), §10.73 and Ch.10

…

19: 19.4 Derivatives and Differential Equations

§19.4 Derivatives and Differential Equations

… ►

§19.4(ii) Differential Equations

►Let $D_k=\partial /\partial k$. Then …If $\varphi =\pi /2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …

20: 9.10 Integrals

… ►

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

ⓘ

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

►

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

ⓘ

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

… ►

9.10.12 $${\int }_{-\mathrm{\infty }}^0\mathrm{Bi}\left(t\right)dt=0.$$

ⓘ

Symbols:

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

Source:

Olver (1997b, p. 431)

Referenced by:

(9.10.7)

Permalink:

http://dlmf.nist.gov/9.10.E12

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

ⓘ

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

…