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Integration with variables Notes_Part_2

# Integration with variables Notes_Part_2 - 1 Re z Figure 7.1...

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Re 1 z Im 1 z Figure 7.1. Real and Imaginary Parts of f ( z ) = 1 z . Therefore, if f ( z ) is any complex function, we can write it as a complex combination f ( z ) = f ( x + i y ) = u ( x, y ) + i v ( x, y ) , of two inter-related real harmonic functions: u ( x, y ) = Re f ( z ) and v ( x, y ) = Im f ( z ). Before delving into the many remarkable properties of complex functions, let us look at some of the most basic examples. In each case, the reader can directly check that the harmonic functions provided by the real and imaginary parts of the complex function are indeed solutions to the Laplace equation. Examples of Complex Functions ( a ) Harmonic Polynomials : As noted above, any complex polynomial is a linear com- bination, as in (7.2), of the basic complex monomials z n = ( x + i y ) n = u n ( x, y ) + i v n ( x, y ) . (7 . 7) Their real and imaginary parts, u n , v n , are the harmonic polynomials that we previously constructed by applying separation of variables to the polar coordinate form of the Laplace equation (4.94). The general formula can be found in (4.110). ( b

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Integration with variables Notes_Part_2 - 1 Re z Figure 7.1...

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