E constant (e = 2.718), Euler's number

E constant or Euler’s number – mathematical constant. The e constant is real and irrational number.

e = 2.718281828459…

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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:

The reciprocal of e is the limit:

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

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_ex)^‘ = (ln x)^‘ = 1/x

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_ex is:

∫ log_ex dx = ∫ lnx dx = x ln x – x + c

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

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

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

ln x = log_ex

The exponential function is defined as:

f(x) = exp(x) = e^x

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.

e constant (e = 2.718)