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1: 32.12 Asymptotic Approximations for Complex Variables

§32.12 Asymptotic Approximations for Complex Variables

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2: 5.1 Special Notation

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$j,m,n$	nonnegative integers.
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$z=x+\mathrm{i}y$	complex variable.
$a,b,q,s,w$	real or complex variables with .
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3: 32.1 Special Notation

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$m,n$	integers.
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$z$	complex variable.
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4: 21.8 Abelian Functions

… ►An Abelian function is a $2g$-fold periodic, meromorphic function of $g$ complex variables. In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

5: 4.34 Derivatives and Differential Equations

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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 of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

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

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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 of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

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

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Annotations for §4.34 and Ch.4

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4.34.12 $$w=(1/a)\sinh \left(az+c\right),$$

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Symbols:

$\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E12

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Annotations for §4.34 and Ch.4

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4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

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Annotations for §4.34 and Ch.4

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4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

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Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

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Annotations for §4.34 and Ch.4

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6: 4.20 Derivatives and Differential Equations

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4.20.9 $$\frac{d^{2}w}{{dz}^2}+a^{2}w=0,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.20.E9

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Annotations for §4.20 and Ch.4

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4.20.10 $${\left(\frac{dw}{dz}\right)}^2+a^2{w}^2=1,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.20.E10

Encodings:

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Annotations for §4.20 and Ch.4

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4.20.12 $$w=A\cos \left(az\right)+B\sin \left(az\right),$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E12

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4.20.13 $$w=(1/a)\sin \left(az+c\right),$$

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Symbols:

$\sin z$: sine function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E13

Encodings:

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4.20.14 $$w=(1/a)\tan \left(az+c\right),$$

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Symbols:

$\tan z$: tangent function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E14

Encodings:

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Annotations for §4.20 and Ch.4

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7: 6.1 Special Notation

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$x$	real variable.
$z$	complex variable.
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8: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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$s=\sigma +\mathrm{i}t$	complex variable.
$z=x+\mathrm{i}y$	complex variable.
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9: 4.14 Definitions and Periodicity

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4.14.4 $$\tan z=\frac{\sin z}{\cos z},$$

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Defines:

$\tan z$: tangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.3

Referenced by:

§4.14, §4.45(i)

Permalink:

http://dlmf.nist.gov/4.14.E4

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4.14.5 $$\csc z=\frac{1}{\sin z},$$

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Defines:

$\csc z$: cosecant function

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

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Annotations for §4.14 and Ch.4

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4.14.6 $$\sec z=\frac{1}{\cos z},$$

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Defines:

$\sec z$: secant function

Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.5

Permalink:

http://dlmf.nist.gov/4.14.E6

Encodings:

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Annotations for §4.14 and Ch.4

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4.14.8 $$\sin \left(z+2k\pi \right)=\sin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.7

Permalink:

http://dlmf.nist.gov/4.14.E8

Encodings:

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Annotations for §4.14 and Ch.4

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4.14.9 $$\cos \left(z+2k\pi \right)=\cos z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.8

Permalink:

http://dlmf.nist.gov/4.14.E9

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Annotations for §4.14 and Ch.4

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10: 4.28 Definitions and Periodicity

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4.28.2 $$\cosh z=\frac{{\mathrm{e}}^z+{\mathrm{e}}^{-z}}{2},$$

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Defines:

$\cosh z$: hyperbolic cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.5.2

Permalink:

http://dlmf.nist.gov/4.28.E2

Encodings:

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Annotations for §4.28 and Ch.4

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4.28.3 $$\cosh z\pm \sinh z={\mathrm{e}}^{\pm z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.19 4.5.20

Permalink:

http://dlmf.nist.gov/4.28.E3

Encodings:

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Annotations for §4.28 and Ch.4

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4.28.5 $$\mathrm{csch}z=\frac{1}{\sinh z},$$

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Defines:

$\mathrm{csch}z$: hyperbolic cosecant function

Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.4

Permalink:

http://dlmf.nist.gov/4.28.E5

Encodings:

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Annotations for §4.28 and Ch.4

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4.28.9 $$\cos \left(\mathrm{i}z\right)=\cosh z,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.8

Permalink:

http://dlmf.nist.gov/4.28.E9

Encodings:

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Annotations for §4.28, §4.28 and Ch.4

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4.28.12 $$\sec \left(\mathrm{i}z\right)=\mathrm{sech}z,$$

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Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function and $z$: complex variable

A&S Ref:

4.5.11

Permalink:

http://dlmf.nist.gov/4.28.E12

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

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