Mathematics MOC

Complex analysis MOC

Complex analysis is the theory of functions of complex variables: an investigation of functions where .

Complex functions fundamentals

• Complex function decomposition
• Singular point (poles, essential singularities)
• Residue

Complex infinitesimal calculus

Many of the rules of real-valued calculus can be applied to complex functions, and many properties can thence be generalised to satisfy the rules of complex numbers. The property has far-reaching consequences.

Differentiability generalises to holomorphic, i.e. analytic.¹ A function is such if and only if it satisfies the Cauchy-Riemann equations.

Complex integration essentially involves computing two path integrals, but as a consequence of the Cauchy-Riemann equations, these are path-independent for any analytic complex function. Cauchy's Integral Formula, Cauchy's Residue Theorem

Series

• Complex functions
• Laurent series

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