E constant or Euler’s number – 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*)

^{‘}= (ln

*x*)

^{‘}= 1/

*x*

**Integrals of e**

The indefinite integral of the exponential function ** e^{x}** is the exponential function

**.**

*e*^{x}*∫ e ^{x} dx = e^{x}+c*

The indefinite integral of the natural logarithm function log*_{e}x* is:

*∫* log_{e}x*dx* = ∫ ln*x dx* = *x* ln *x – x + c*

The definite integral from 1 to * e* of the reciprocal function

**1/**

*is 1:*

**x**### Base e logarithm

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

*logarithm of*

**e***:*

**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 * e^{iθ}* has the identity:

*e ^{iθ}* = cos(

*θ*) +

*i*sin(

*θ*)

* i* is the imaginary unit (the square root of -1).

*θ* – any real number.