app1 - Poularikas A.D Appendix 1 Functions of a Complex...

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Poularikas, A.D . Appendix 1: Functions of a Complex Variable .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000

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© 2000 by CRC Press LLC Appendix 1: Functions of a Complex Variable * Alexander D. Poularikas University of Alabama in Huntsville 1 Basic Concepts A complex variable z defined by z = x + jy (1.1) assumes certain values over a region R z of the complex plane. If a complex quantity W ( z ) is so connected with z that each z in R z corresponds with one value of W ( z ) in R w , then we say that W ( z ) is a single- valued function of z W ( z ) = u ( x, y ) + j υ ( x, y ) (1.2) which has a domain R z and a range R w (see Figure 1.1 ). The function W ( z ) can be single valued or multiple valued. Examples of single-valued functions include W = a 0 + a 1 z + a 2 z 2 + L + a n z n n integer W = e z Examples of multiple-valued functions are W = z n n not an integer W = log z W = sin –1 z DEFINITION 1.1 A function W ( z ) is continuous at a point z = λ of R z if, for each number ε > 0, however small, there exists another number δ > 0 such that whenever z λ < δ then W ( z ) – W ( λ ) < ε (1.3) The geometric representation of this equation is shown in Figure 1.1 . * All contour integrals are taken counterclockwise, unless specifically indicated.
© 2000 by CRC Press LLC DEFINITION 1.2 A function W(z) is analytic at a point z if, for each number ε > 0, however small, there exist another number δ > 0 such that whenever (1.4) Example 1.1 Show that the function W ( z ) = e z satisfies (1.4). Solution From (1.4), we obtain which proves the assertion. In this example, we did not mention the direction from which the z approaches λ . We might surmise from this that the derivative of our analytic function is independent of the path of z as it approaches the limiting point. However, this is not true in general. By setting λ = z and z = z + z in (1.4), we obtain an alternative form of that equation, namely, (1.5) For a function to possess a unique derivative, it is required that FIGURE 1.1 Illustration of the range and domain of complex functions. z then W z W z dW dz λ λ − λ λ < ( ) ( ) ( ) < δ ε lim lim lim ! ! z z z z z z z W z W z e e z e z z e de dz = ( ) ( ) = = ( ) + ( ) = = λ λ λ λ λ λ λ λ λ λ λ 1 2 3 2 L dW dz W z z W z z z = + ( ) ( ) lim 0

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© 2000 by CRC Press LLC But because the unique derivative becomes For this to be independent of how x and y approach zero (that is, for the derivative to be unique), it is necessary and sufficient that x + j y cancel in the numerator and denominator. This requires that This condition can be met if (1.6) These are the Cauchy-Riemann conditions. If the function satisfies these equations, it possesses a unique derivative and it is analytic at that point. These conditions are necessary and sufficient.
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