514 use theorem 44 lecture 6 analytic functions i in

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5.14. Use Theorem 4.4.
Lecture 6 Analytic Functions I In this lecture, using the fundamental notion of limit, we shall define the di ff erentiation of complex functions. This leads to a special class of functions known as analytic functions. These functions are of great im- portance in theory as well as applications, and constitute a major part of complex analysis. We shall also develop the Cauchy-Riemann equations which provide an easier test to verify the analyticity of a function. Let f be a function defined in a neighborhood of a point z 0 . Then, the derivative of f at z 0 is given by df dz ( z 0 ) = f ( z 0 ) = lim z 0 f ( z 0 + z ) f ( z 0 ) z , (6 . 1) provided this limit exists. Such a function f is said to be di ff erentiable at the point z 0 . Alternatively, f is di ff erentiable at z 0 if and only if it can be written as f ( z ) = f ( z 0 ) + A ( z z 0 ) + η ( z )( z z 0 ); (6 . 2) here, A = f ( z 0 ) and η ( z ) 0 as z z 0 . Clearly, in (6.1), z can go to zero in infinite di ff erent ways. Example 6.1. Show that, for any positive integer n, d dz z n = nz n 1 . Using the binomial formula, we find ( z + z ) n z n z = n 1 z n 1 z + n 2 z n 2 ( z ) 2 + · · · + n n ( z ) n z = nz n 1 + n ( n 1) 2 z n 2 z + · · · + ( z ) n 1 . Thus, d dz z n = lim z 0 ( z + z ) n z n z = nz n 1 . Example 6.2. Clearly, the function f ( z ) = z is continuous for all z. We shall show that it is nowhere di ff erentiable. Since f ( z 0 + z ) f ( z 0 ) z = ( z 0 + z ) z 0 z = z z . R.P. Agarwal et al., An Introduction to Complex Analysis , DOI 10.1007/978-1-4614-0195-7_6, © Springer Science+Business Media, LLC 2011 37
38 Lecture 6 If z is real, then z = z and the di ff erence quotient is 1 . If z is purely imaginary, then z = z and the quotient is 1 . Hence, the limit does not exist as z 0 . Thus, z is not di ff erentiable. In real analysis, construction of functions that are continuous everywhere but di ff erentiable nowhere is hard. The proof of the following results is almost the same as in calculus. Theorem 6.1. If f and g are di ff erentiable at a point z 0 , then (i). ( f ± g ) ( z 0 ) = f ( z 0 ) ± g ( z 0 ) , (ii). ( cf ) ( z 0 ) = cf ( z 0 ) ( c is a constant), (iii). ( fg ) ( z 0 ) = f ( z 0 ) g ( z 0 ) + f ( z 0 ) g ( z 0 ) , (iv). f g ( z 0 ) = g ( z 0 ) f ( z 0 ) f ( z 0 ) g ( z 0 ) ( g ( z 0 )) 2 if g ( z 0 ) ̸ = 0 , and (v). ( f g ) ( z 0 ) = f ( g ( z 0 )) g ( z 0 ) , provided f is di ff erentiable at g ( z 0 ) . Theorem 6.2. If f is di ff erentiable at a point z 0 , then f is continuous at z 0 . A function f of a complex variable is said to be analytic (or holomorphic , or regular ) in an open set S if it has a derivative at every point of S. If S is not an open set, then we say f is analytic in S if f is analytic in an open set containing S. We call f analytic at the point z 0 if f is analytic in some neighborhood of z 0 . It is important to note that while di ff erentiability is defined at a point, analyticity is defined on an open set. If a function f is analytic on the whole complex plane, then it is said to be entire .

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