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Sometimes, as in the case of the natural logarithm , it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.

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All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension in which the analytic properties such as power series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension such as conformality do not carry over. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.

A major use of certain complex spaces is in quantum mechanics as wave functions.

Introduction to Complex Analysis

From Wikipedia, the free encyclopedia. Branch of mathematics studying functions of a complex variable. For the class of functions often called "complex analytic", see Holomorphic function.

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Not to be confused with Complexity theory. Main article: Holomorphic function. Complex analysis at Wikipedia's sister projects. Categories : Complex analysis.

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By using this site, you agree to the Terms of Use and Privacy Policy. Real number Imaginary number Complex plane Complex conjugate Unit complex number. Complex-valued function Analytic function Holomorphic function Cauchy—Riemann equations Formal power series.

Complex Analysis

Zeros and poles Cauchy's integral theorem Local primitive Cauchy's integral formula Winding number Laurent series Isolated singularity Residue theorem Conformal map Schwarz lemma Harmonic function Laplace's equation. Mathematics portal. Article no.

Mod-02 Lec-01 Introduction to Complex Functions

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