Math 2.0

Special functions Theorem

Product of exponentials

Builds on Complex numbers

Statement

For all complex numbers zz and ww, ez+w=ezew,e^{z + w} = e^{z} e^{w}, where ez=∑n=0∞zn/n!e^{z} = \sum_{n=0}^{\infty} z^n / n! is the complex exponential function.

Remarks

The proof multiplies the two series and uses the binomial theorem. It follows that ez≠0e^{z} \neq 0 and e−z=1/eze^{-z} = 1 / e^{z} for each zz. With Euler's formula, it gives the addition formulas for cos⁡\cos and sin⁡\sin.

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Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Functional equation
  2. Web https://en.wikipedia.org/wiki/Exponential_function, Complex exponential
About this page
Tag0WM7DJCM6G
URIm2://0WM7DJCM6G
Aliasesexponential-of-a-sum
Kindstatement, theorem
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 0WM7DJCM6G.
UsesComplex numbers

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