section_path: Complex analysis/Complex exponential
section_name: Complex exponential
paragraph_text: Complex exponential
  
  Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp z, also written e, is defined as the infinite series, which can be shown to converge for any z: exp z:= 1 + z + z 2 2 ⋅ 1 + z 3 3 ⋅ 2 ⋅ 1 + ⋯ = ∑ n = 0 ∞ z n n !. {\displaystyle \exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.} For example, exp (1) {\displaystyle \exp(1)} is Euler's constant e ≈ 2.718 {\displaystyle e\approx 2.718}. Euler's formula states: exp (i φ) = cos φ + i sin φ {\displaystyle \exp(i\varphi)=\cos \varphi +i\sin \varphi } for any real number φ. This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes Euler's identity exp (i π) = − 1. {\displaystyle \exp(i\pi)=-1.}