By Jan G. Krzyz
ISBN-10: 0444000984
ISBN-13: 9780444000989
Read Online or Download Problems in Complex Variable Theory PDF
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Extra info for Problems in Complex Variable Theory
Sample text
Hint: Consider the function t h(t) = (' ~iLdT J z(T)-a '" where z = z(t), 0( ~ t ~ (3, is the equation of y. Verify that u(t) -a] exp(-h(t)) is a constant. 20. Show that n(y, a) has a constant value for all a ED, if the domain D does not contain any points of y. 21. 1. Verify that n(y, a) a EC",L1. 22. Suppose I' is a regular, closed curve not meeting the negative real axis. Show that for any a E ( - 00, 0) we have n(y, a) = O. 23. t. the origin: If I' has the equation z(O) = r(O)e i9 , 0 ~ 0 ~ 27t, r(O) = r(27t) and r(O) is a positive, continuously differentiable function of 0,' then C", {I'} = Do u D oo , where Do, Doo are disjoint domains with a common boundary y.
Any pair of conjugate harmonic functions u,v determines an analytic function u+iv. 1. Find all the functions harmonic 10 C"", (- 00, 0] which are cOllstant one the rays argz = const. 2. Find all the functions harmonic in C"", 0 which are constant on the circles ceO; r). 3. Verify that the functions u = log Izl, v = argz are conjugate harmonic functions in C"", (-00,0] and Logz = log Izl+iArgz, where Argz is the principal value of argument: -7t < Argz < 7t, is analytic in C"",(- 00, 0]. 4. Verify that ~cosy, ~siny are conjugate harmonic functions in C.
T. t. 1. Find the image domain of the linear transformation: W= (i) {z: rez > 0, imz > O}, W = (ii) {z: Izl < 1, imz > O}, W= (iii) {z: 0 < argz < -t7t}, W= (iv) {z: 0 < rez < 1}, W= (v) {z: 1 < Izl < 2}, given domain in z-plane under the given (z-i) (Z+i)-l; (2z-i) (2+iz)-1; z(z-lrl; (z-l) (Z-2)-1; z(Z-l)-l. 2. Find the linear transformation carrying the circle C(O; 1) into a straight line parallel to the imaginary axis, the point z = 4 into the point W = 0 and leaving the circle C(O; 2) invariant.