e constant

e constant or Euler's number is 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:

e=\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^x = 2.718281828459...

Alternative definitions

The e constant is defined as the limit:

e=\lim_{x\rightarrow 0 }\left ( 1+ \right x)^\frac{1}{x}

 

The e constant is defined as the infinite series:

e=\sum_{n=0}^{\infty }\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...

Properties of e

Reciprocal of e

The reciprocal of e is the limit:

\lim_{x\rightarrow \infty }\left ( 1-\frac{1}{x} \right )^x=\frac{1}{e}

Derivatives of e

The derivative of the exponential function is the exponential function:

(e x)' = ex

The derivative of the natural logarithm function is the reciprocal function:

(loge x)' = (ln x)' = 1/x

 

Integrals of e

The indefinite integral of the exponential function ex is the exponential function ex.

ex dx = ex+c

 

The indefinite integral of the natural logarithm function loge x is:

∫ loge x dx = ∫ ln x dx = x ln x - x +c

 

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

\int_{1}^{e}\frac{1}{x}\: dx=1

 

Base e logarithm

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

ln x = loge x

Exponential function

The exponential function is defined as:

f (x) = exp(x) = ex

Euler's formula

The complex number e has the identity:

e = cos(θ) + i sin(θ)

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

θ is any real number.

 


See also

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