# exponential function

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##### 1: 4.2 Definitions
###### §4.2(iii) The ExponentialFunction
4.2.19 $\exp z=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots.$
The function $\exp$ is an entire function of $z$, with no real or complex zeros. …
4.2.24 $\exp z=e^{x}\cos y+ie^{x}\sin y.$
##### 3: 6.11 Relations to Other Functions
###### Incomplete Gamma Function
6.11.1 $E_{1}\left(z\right)=\Gamma\left(0,z\right).$
##### 5: 17.17 Physical Applications
See Kassel (1995). …
##### 7: 4.1 Special Notation
 $k,m,n$ integers. …
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 $\ln z$, $\operatorname{Ln}z$; the exponential $\exp z$, $e^{z}$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\operatorname{arcsin}z$, $\operatorname{Arcsin}z$, etc. …
##### 8: 4.3 Graphics Figure 4.3.1: ln ⁡ x and e x . … Magnify
###### §4.3(ii) Complex Arguments: Conformal Maps Figure 4.3.4: e x + i ⁢ y . Magnify 3D Help
##### 9: 4.12 Generalized Logarithms and Exponentials
A generalized exponential function $\phi(x)$ satisfies the equations …
4.12.6 $\phi(x)=\ln\left(x+1\right),$ $-1,
4.12.7 $\phi(x)=\underbrace{\exp\cdots\exp}_{\left\lfloor x\right\rfloor\text{ times}}% (x-\left\lfloor x\right\rfloor),$ $x>1$.
##### 10: 17.3 $q$-Elementary and $q$-Special Functions
###### $q$-ExponentialFunctions
17.3.1 $e_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^{n}}{\left(q;q\right)_% {n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$
17.3.2 $E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$
17.3.4 $\mathrm{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2n+1}}{% \left(q;q\right)_{2n+1}}.$
17.3.6 $\mathrm{Cos}_{q}\left(x\right)=\frac{1}{2}(E_{q}\left(ix\right)+E_{q}\left(-ix% \right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}}{\left(q% ;q\right)_{2n}}.$