exponential function

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1: 4.2 Definitions

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§4.2(iii) The Exponential Function

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4.2.19 $$\exp z=1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\mathrm{\cdots }.$$

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Defines:

$\exp z$: exponential function

Symbols:

$!$: factorial (as in $n!$) and $z$: complex variable

A&S Ref:

4.2.1

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.2.E19

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Annotations for §4.2(iii), §4.2 and Ch.4

►The function $\exp$ is an entire function of $z$, with no real or complex zeros. … ►

4.2.24 $$\exp z={\mathrm{e}}^x\cos y+\mathrm{i}{\mathrm{e}}^x\sin y.$$

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Symbols:

$\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E24

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Annotations for §4.2(iii), §4.2 and Ch.4

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4.2.32 $${\mathrm{e}}^z=\exp z,$$

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Symbols:

$\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E32

Encodings:

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Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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2: 4.11 Sums

§4.11 Sums

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3: 6.11 Relations to Other Functions

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Incomplete Gamma Function

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6.11.1 $$E_1\left(z\right)=\mathrm{\Gamma }(0,z).$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

5.1.45 (special case of)

Referenced by:

§6.11

Permalink:

http://dlmf.nist.gov/6.11.E1

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Annotations for §6.11, §6.11 and Ch.6

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Confluent Hypergeometric Function

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

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Annotations for §6.11, §6.11 and Ch.6

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4: 4.47 Approximations

§4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

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5: 17.17 Physical Applications

… ►See Kassel (1995). …

6: 4.46 Tables

§4.46 Tables

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7: 4.1 Special Notation

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$k,m,n$	integers.
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… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. …

8: 4.3 Graphics

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. … Magnify

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§4.3(ii) Complex Arguments: Conformal Maps

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Figure 4.3.4: ${\mathrm{e}}^{x+\mathrm{i}y}$. Magnify 3D Help

9: 8.5 Confluent Hypergeometric Representations

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8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

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Annotations for §8.5 and Ch.8

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8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

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Annotations for §8.5 and Ch.8

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8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

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Annotations for §8.5 and Ch.8

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8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

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Annotations for §8.5 and Ch.8

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8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

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Annotations for §8.5 and Ch.8

10: 4.12 Generalized Logarithms and Exponentials

… ►A generalized exponential function $\varphi (x)$ satisfies the equations … ►

4.12.6 $$\varphi (x)=\ln \left(x+1\right),$$ ,

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Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E6

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Annotations for §4.12 and Ch.4

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4.12.7 $$\varphi (x)=\underset{\lfloor x\rfloor \text{ times}}{\underbrace{\exp \mathrm{\cdots }\exp }}(x-\lfloor x\rfloor ),$$ $x>1$.

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Symbols:

$\exp z$: exponential function, $\lfloor x\rfloor$: floor of $x$ , $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E7

Encodings:

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Annotations for §4.12 and Ch.4

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