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MATH
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Theorems from Trigonometric and geometric applications

All theorems from Trigonometric and geometric applications.

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Binomial theorem over the complex numbers

For a,b∈Ca,b\in\mathbb C and every non-negative integer nn,

(a+b)n=∑k=0n(nk)an−kbk,(nk)=n!k!(n−k)!.(a+b)^n=\sum_{k=0}^n{n\choose k}a^{n-k}b^k, \qquad {n\choose k}=\frac{n!}{k!(n-k)!}.

The theorem holds over C\mathbb C because complex addition and multiplication obey the same commutative and distributive laws as real arithmetic. The coefficients can be read from Pascal's triangle.

  1. [1]:Complex number
  2. [2]:Expand powers of sine and cosine

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Binomial theorem over the complex numbers