holomorphic differentials

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1: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

…

2: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

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15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

Encodings:

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See also:

Annotations for §15.10(i), §15.10 and Ch.15

►This is the hypergeometric differential equation. … ► …

3: 21.7 Riemann Surfaces

… ►On a Riemann surface of genus $g$, there are $g$ linearly independent holomorphic differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$. If a local coordinate $z$ is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by …Thus the differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$ have no singularities on $\mathrm{\Gamma }$. Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. … ►Define the holomorphic differential …

4: 1.9 Calculus of a Complex Variable

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${\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{\partial v}{\partial y}},$

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${\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{\partial v}{\partial x}}$

… ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. ►

Analyticity

►A function $f(z)$ is said to be analytic (holomorphic) at $z=z_0$ if it is complex differentiable in a neighborhood of $z_0$. …

5: 1.5 Calculus of Two or More Variables

… ►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, and ${\partial }^{2}f/\partial y\partial x$ are continuous. … ►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime }(x)=-F_x/F_y$. … ►Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$. … ►Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty }$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. Suppose also that ${\int }_c^{d}f(x,y)dy$ converges and ${\int }_c^d(\partial f/\partial x)dy$ converges uniformly on $a\le x\le b$, that is, given any positive number $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that …

6: 36.10 Differential Equations

§36.10 Differential Equations

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36.10.3 $$\frac{{\partial }^2{\mathrm{\Psi }}_1}{{\partial x}^2}-\frac{x}{3}{\mathrm{\Psi }}_1=0.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/36.10.E3

Encodings:

pMML, png, TeX

See also:

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

… ► $K=2$, cusp: … ► $K=3$, swallowtail: … ►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations …

7: 19.18 Derivatives and Differential Equations

§19.18 Derivatives and Differential Equations

… ►Let ${\partial }_j=\partial /\partial z_j$, and ${𝐞}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … ►

§19.18(ii) Differential Equations

… ►and also a system of $n(n-1)/2$ Euler–Poisson differential equations (of which only $n-1$ are independent): …If $n=2$, then elimination of ${\partial }_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_1,b_2,z_1,z_2)=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). …

8: 16.14 Partial Differential Equations

§16.14 Partial Differential Equations

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§16.14(i) Appell Functions

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$$x(1-x)\frac{{\partial }^2{F}_1}{{\partial x}^2}+y(1-x)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_1}{\partial x}-\beta y\frac{\partial F_1}{\partial y}-\alpha \beta F_1=0,$$

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$$y(1-y)\frac{{\partial }^2{F}_1}{{\partial y}^2}+x(1-y)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +{\beta }^{\prime }+1)y\right)\frac{\partial F_1}{\partial y}-{\beta }^{\prime }x\frac{\partial F_1}{\partial x}-\alpha {\beta }^{\prime }{F}_1=0,$$

… ►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …

9: 4.10 Integrals

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4.10.1 $$\int \frac{dz}{z}=\ln z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.48

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.2 $$\int \ln zdz=z\ln z-z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.49

Permalink:

http://dlmf.nist.gov/4.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►

4.10.4 $$\int \frac{dz}{z\ln z}=\ln \left(\ln z\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.52

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.5 $${\int }_0^1\frac{\ln t}{1-t}dt=-\frac{{\pi }^2}{6},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.55

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.6 $${\int }_0^1\frac{\ln t}{1+t}dt=-\frac{{\pi }^2}{12},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.56

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

…

10: 10.38 Derivatives with Respect to Order

… ►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:

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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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${{\displaystyle \frac{\partial K_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=0.$

… ►

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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