6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that Re /(z)- Re g(z) for all z e B. Show that J-g + ία in D, whe...
8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D. Show that if f is never zero on D, then the minimum of lfl is assumed on B. You will need to use the fact that lfl does, indeed, assume a minimum somewhere on D B 8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D....
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.) 12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
(b) Suppose f is analytic in a neighborhood of the closed and bounded nonempty domain D. Assuming that f(z) # 0 for all z e D, show that the minimum value of \f| is achieved on the boundary ƏD of D. (Bonus: can you give a contradiction to the above statement if you assume f vanishes at a point in the interior of D?)
11. (8)(a) Suppose that f and g are analytic branches of zt on a domain D such that 0 g D Show that there is a fifth root, wo, of 1 such that f(z)-wog(2) for all E D. I suggest considering h(z) f (z)/g(z) (b) Now suppose that D D C(-,0]. Let f be an analytic branch of zt in D such that f (1) 1. Show that f(z) expLog(2)) for all z ED. 11. (8)(a) Suppose that f and...
D. (a) Show that if f is Let f be complex differentiable on a bounded domain D and is continuous on the closure D = DU non-zero on D the modulus $(2) attains it's minimum on the boundary aD. Hint: Consider FT2) (b) Give an example that shows that the assumption that f is non-zero on D is necessary.
f(D) CL/ D. If either 310 pts]. Let f be an analytic function defined on a domain or f(D) C C where f (D) denotes the range of f, L is any straight line and C is any circle in C, then show that f must be constant in D. f(D) CL/ D. If either 310 pts]. Let f be an analytic function defined on a domain or f(D) C C where f (D) denotes the range of f, L...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the origin. Then do the following ones: (a) Show that rurr(r, θ) + rur(r, θ) + u69(r, θ) 0 for all re® E D. (b) Show that (a) is equivalent to the condition that u is harmonic in D (c) Show that the function (in|e )2-[Arg( a(z) z)]2,-π < Arg(z) < π, 4[10 pts]. Let f(z) = u (r,0)...
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
complex analysis Let f(z) be continuous on S where for some real numbers 0< a < b. Define max(Re(z)Im(z and suppose f(z) dz = 0 S, for all r E (a, b). Prove or disprove that f(z) is holomorphic on S.