Math 2.0

Special functions Definition

Complex exponential

Builds on Complex numbers

Definition

The complex exponential is the function exp⁡:ℂ→ℂ\exp : \mathbb{C} \to \mathbb{C} given by exp⁡z=∑n=0∞znn!=1+z+z22!+z33!+⋯\exp z = \sum_{n=0}^{\infty} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots for each complex number zz. The series converges absolutely for each z∈ℂz \in \mathbb{C}. Another notation for exp⁡z\exp z is eze^z.

Remarks

For real xx, exp⁡x\exp x is the real exponential function, and e=exp⁡1e = \exp 1. For z=x+iyz = x + iy with x,yx, y real, ez=ex(cos⁡y+isin⁡y)e^z = e^x (\cos y + i \sin y). The case x=0x = 0 is Euler's formula. By the product of exponentials, ez+w=ezewe^{z + w} = e^z e^w. So ez≠0e^z \neq 0 for each zz, and ∣ez∣=eRe⁡z|e^z| = e^{\operatorname{Re} z}.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Complex exponential
  2. Web https://en.wikipedia.org/wiki/Exponential_function, Power series
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