complex2 - 2 Complex Functions and the Cauchy-Riemann...

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Unformatted text preview: 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f ( x ) of a real variable x . Like- wise, in complex analysis, we study functions f ( z ) of a complex variable z C (or in some region of C ). Here we expect that f ( z ) will in general take values in C as well. However, it will turn out that some functions are better than others. Basic examples of functions f ( z ) that we have already seen are: f ( z ) = c , where c is a constant (allowed to be complex), f ( z ) = z , f ( z ) = z , f ( z ) = Re z , f ( z ) = Im z , f ( z ) = | z | , f ( z ) = e z . The func- tions f ( z ) = arg z , f ( z ) = z , and f ( z ) = log z are also quite interesting, but they are not well-defined (single-valued, in the terminology of complex analysis). What is a complex valued function of a complex variable? If z = x + iy , then a function f ( z ) is simply a function F ( x, y ) = u ( x, y ) + iv ( x, y ) of the two real variables x and y . As such, it is a function (mapping) from R 2 to R 2 . For example, f ( z ) = z corresponds to F ( x, y ) = x + iy ; f ( z ) = z to F ( x, y ) = x- iy , f ( z ) = | z | to F ( x, y ) = p x 2 + y 2 . Here, this last example takes values just along the real axis. If f ( z ) = u + iv , then the function u ( x, y ) is called the real part of f and v ( x, y ) is called the imaginary part of f . Of course, it will not in general be possible to plot the graph of f ( z ), which will lie in C 2 , the set of ordered pairs of complex numbers, but it is the set { ( z, w ) C 2 : w = f ( z ) } . The graph can also be viewed as the subset of R 4 given by { ( x, y, s, t ) : s = u ( x, y ) , t = v ( x, y ) } . In particular, it lies in a four-dimensional space. The usual operations on complex numbers extend to complex functions: given a complex function f ( z ) = u + iv , we can define functions Re f ( z ) = u , Im f ( z ) = v , f ( z ) = u- iv , | f ( z ) | = u 2 + v 2 . Likewise, if g ( z ) is another complex function, we can define f ( z ) g ( z ) and f ( z ) /g ( z ) for those z for which g ( z ) 6 = 0. Some of the most interesting examples come by using the algebraic op- erations of C . For example, a polynomial is an expression of the form P ( z ) = a n z n + a n- 1 z n- 1 + + a , where the a i are complex numbers, and it defines a function in the usual way. It is easy to see that the real and imaginary parts of a polynomial P ( z ) are polynomials in x and y . For example, P ( z ) = z 2 = x 2- y 2 + 2 xyi. 1 But given two (real) polynomial functions u ( x, y ) and z ( x, y ), it is very rarely the case that there exists a complex polynomial P ( z ) such that P ( z ) = u + iv ....
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This note was uploaded on 10/18/2011 for the course MATH S1202Q taught by Professor Peters during the Spring '11 term at Columbia.

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complex2 - 2 Complex Functions and the Cauchy-Riemann...

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