# Exponential

The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula
$$\exp(z) \equiv e^z = \sum_{k=0}^{\infty} \dfrac{z^k}{k!},$$
where $e$ is the base of the natural logarithm.

Domain coloring of $\exp$.

# Properties

Derivative of the exponential function

Taylor series of the exponential function

Euler E generating function

Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt

Exponential function is periodic with period 2pii

Euler's formula

E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1

E^x is greater than 1+x for nonzero real x

E^x is less than 1/(1-x) for nonzero real x less than 1

X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1

X less than e^x-1 less than x/(1-x) for nonzero real x less than 1

1+x greater than exp(x/(1+x)) for nonzero real x greater than -1

E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0

E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)

E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936

Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1

Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))

Sum of cosh and sinh

Difference of cosh and sinh

## References

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.2.1$