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Methods of Mathematical Physics, Third Edition by Bertha Jeffreys, Harold Jeffreys

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Chapter 11FUNCTIONS OF A COMPLEX VARIABLE

Of fowls after their kind, and of cattle after their kind, of every creeping thing of the earth after his kind, two of every sort shall come unto thee, to keep them alive.

Genesis vi, 20

11·01. Meaning and algebra of complex numbers.

There are three chief reasons why complex functions, involving a symbol i such that i.i = –1, are of importance in physics, which involves only real quantities directly. The first is that many physical quantities are functions ϕ, ψ of two variables x and y, where ϕ and ψ are connected by the relations

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Such pairs of functions occur, for instance, in two-dimensional problems ...

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