## e constant

e constantorEuler's numberis a mathematical constant. The e constant is real and irrational number.

e= 2.718281828459...## Definition of e

The e constant is defined as the limit:

## Alternative definitions

The e constant is defined as the limit:

The e constant is defined as the infinite series:

## Properties of e

## Reciprocal of e

The reciprocal of e is the limit:

## Derivatives of e

The derivative of the exponential function is the exponential function:

(

e)' =^{ x}e^{x}The derivative of the natural logarithm function is the reciprocal function:

(log

)_{e }x' =(lnx)'=1/x

## Integrals of e

The indefinite integral of the exponential function e

^{x}is the exponential function e^{x}.∫

e=^{x }dxe+c^{x}

The indefinite integral of the natural logarithm function log

is:_{e }x∫ log

= ∫ ln_{e }x dx=_{ }x dxxlnx - x+c

The definite integral from 1 to e of the reciprocal function 1/x is 1:

## Base e logarithm

The natural logarithm of a number x is defined as the base e logarithm of x:

ln

x= log_{e }x## Exponential function

The exponential function is defined as:

f(x) = exp(x) =e^{x}## Euler's formula

The complex number

ehas the identity:^{ iθ}

e= cos(^{iθ}θ) +isin(θ)i is the imaginary unit (the square root of -1).

θ is any real number.

## See also