section_path: Properties/Complex numbers
section_name: Complex numbers
paragraph_text: Complex numbers
  
  The exponential function e may be written as a Taylor series
  
  e x = 1 + x 1 ! + x 2 2 ! + x 3 3 ! + ⋯ = ∑ n = 0 ∞ x n n !. {\displaystyle e^{x}=1+{x \over 1!}+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}
  
  Because this series is convergent for every complex value of x, it is commonly used to extend the definition of e to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:
  
  e i x = cos x + i sin x, {\displaystyle e^{ix}=\cos x+i\sin x,}
  
  which holds for every complex x. The special case with x = π is Euler's identity:
  
  e i π + 1 = 0, {\displaystyle e^{i\pi }+1=0,} which is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle. Moreover, the identity implies that, in the principal branch of the logarithm,
  
  ln (− 1) = i π. {\displaystyle \ln(-1)=i\pi.}
  
  Furthermore, using the laws for exponentiation,
  
  (cos x + i sin x) n = (e i x) n = e i n x = cos n x + i sin n x {\displaystyle (\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos nx+i\sin nx}
  
  for any integer n, which is de Moivre's formula.
  
  The expressions of cos x and sin x in terms of the exponential function can be deduced from the Taylor series: cos x = e i x + e − i x 2, sin x = e i x − e − i x 2 i. {\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \sin x={\frac {e^{ix}-e^{-ix}}{2i}}.}
  
  The expression cos x + i sin x {\textstyle \cos x+i\sin x} is sometimes abbreviated as cis(x).