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1: 4.20 Derivatives and Differential Equations

§4.20 Derivatives and Differential Equations

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4.20.1 $$\frac{d}{dz}\sin z=\cos z,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.105

Permalink:

http://dlmf.nist.gov/4.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.2 $$\frac{d}{dz}\cos z=-\sin z,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.106

Permalink:

http://dlmf.nist.gov/4.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.3 $$\frac{d}{dz}\tan z={\sec }^{2}z,$$

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Symbols:

$\frac{df}{dx}$: derivative, $\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.107

Permalink:

http://dlmf.nist.gov/4.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.4 $$\frac{d}{dz}\csc z=-\csc z\cot z,$$

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Symbols:

$\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative and $z$: complex variable

A&S Ref:

4.3.108

Permalink:

http://dlmf.nist.gov/4.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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2: 4.34 Derivatives and Differential Equations

§4.34 Derivatives and Differential Equations

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4.34.1 $$\frac{d}{dz}\sinh z=\cosh z,$$

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Symbols:

$\frac{df}{dx}$: derivative, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.71

Permalink:

http://dlmf.nist.gov/4.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.2 $$\frac{d}{dz}\cosh z=\sinh z,$$

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Symbols:

$\frac{df}{dx}$: derivative, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.72

Permalink:

http://dlmf.nist.gov/4.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.7 $$\frac{d^{2}w}{{dz}^2}-a^{2}w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative, $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.8 $${\left(\frac{dw}{dz}\right)}^2-a^2{w}^2=1,$$

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Symbols:

$\frac{df}{dx}$: derivative, $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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3: 4.7 Derivatives and Differential Equations

§4.7 Derivatives and Differential Equations

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§4.7(i) Logarithms

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4.7.1 $$\frac{d}{dz}\ln z=\frac{1}{z},$$

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Symbols:

$\frac{df}{dx}$: derivative, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.46

Permalink:

http://dlmf.nist.gov/4.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

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§4.7(ii) Exponentials and Powers

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4.7.12 $$\frac{dw}{dz}=f(z)w$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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4: 22.13 Derivatives and Differential Equations

§22.13 Derivatives and Differential Equations

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§22.13(i) Derivatives

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Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

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$\frac{d}{dz}(\mathrm{sn}z)$ $=$	$\mathrm{cn}z\mathrm{dn}z$	$\frac{d}{dz}(\mathrm{dc}z)$ =	$k_{}^{\prime }{}_{}{}^2\mathrm{sc}z\mathrm{nc}z$
…

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►Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. … ►

22.13.1 $${\left(\frac{d}{dz}\mathrm{sn}(z,k)\right)}^2=\left(1-{\mathrm{sn}}^2(z,k)\right)\left(1-k^2{\mathrm{sn}}^2(z,k)\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $z$: complex and $k$: modulus

Referenced by:

§22.13(iii), §22.18(iv)

Permalink:

http://dlmf.nist.gov/22.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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5: 7.10 Derivatives

§7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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${\displaystyle \frac{d\mathrm{f}\left(z\right)}{dz}}=-\pi z\mathrm{g}\left(z\right),$

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${\displaystyle \frac{d\mathrm{g}\left(z\right)}{dz}}=\pi z\mathrm{f}\left(z\right)-1.$

6: 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, $\partial x$: partial differential, $y$: 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, $\partial x$: partial differential, $y$: 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, $\partial x$: partial differential, $y$: 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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7: 1.5 Calculus of Two or More Variables

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§1.5(i) Partial Derivatives

… ►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. … ►

Chain Rule

… ►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 $[a,b]\times [c,d)$. … ►

§1.5(vi) Jacobians and Change of Variables

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8: 14.29 Generalizations

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14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

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9: 19.18 Derivatives and Differential Equations

§19.18 Derivatives and Differential Equations

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§19.18(i) Derivatives

… ►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. … ►

19.18.14 $$\frac{{\partial }^{2}w}{{\partial x}^2}=\frac{{\partial }^{2}w}{{\partial y}^2}+\frac{1}{y}\frac{\partial w}{\partial y}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/19.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

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19.18.15 $$\frac{{\partial }^{2}W}{{\partial t}^2}=\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential and $W$: wave function

Permalink:

http://dlmf.nist.gov/19.18.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

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10: 30.12 Generalized and Coulomb Spheroidal Functions

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30.12.1 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +\alpha z+{\gamma }^2(1-z^2)-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.12

Permalink:

http://dlmf.nist.gov/30.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

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30.12.2 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +{\gamma }^2(1-z^2)-\frac{\alpha (\alpha +1)}{z^2}-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Permalink:

http://dlmf.nist.gov/30.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

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