Problem 4. (5 points) Suppose f is analytic on and inside a simple closed curve C....
9. Suppose that f (z) has a simple pole at ao on a closed curve C, but is analytic elsewhere inside and on C except for poles at a finite number of interior points a1,a2,, (a) If the contour C is indented at ao by a circular arc with center at ao, show that the limiting form of the integral of f (x) around the indented contour is as the radius of the indentation tends to zero, regardless of whether...
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed half plane H = {2€ C: Im(x) > 0} and that there is a finite constant M with f() < M for all z H. 1. Show that da = f(i) x² +1 +00 2. Show that if o is a point in C with Im(a) > 0, then I (a) Im(a)' 22-2Re(a)x+ lajar (3) deduce sin (Bx) where 870
(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?)
Exercise 4 (15 points) (a) Calculate endz where C is a close simple counter-clockwise curve shown in Figure 4. (15 points) Hint: The function f(z) e is analytic. (you do not have to prove the function is analytic) Solution to Exercise 4 1 2 3 4 -1 Figure 4
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).)
(20 points) Let and let C' be any simple closed curve in a plane oriented counterclockwise. Please show that the only two possible values for F. dr is 0 or-2π. (Hint) The domain of the vector field does not include the origin. Hence, the origin is seen as a hole. Consider 1) Curve C does not encompass the origin. 2) Curve C does encompass the origin. In this case, use an auxiliary curve that encompasses the origin and is encompassed...
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...
4. Suppose (x) (3+4x)+e*. a) Use analytic methods to show f (x is one to one. b) Find () (28) Suppose g (x)--fx-12 + 3e 30-1 5x + e 2-x-8x + 17 . ₩5. c) Use analytic methods to show g(x) is one to one. d) Find (g") (4) 3x-2 6. Find the equation of the tangent line to the curve y -at the po int (Q,e)
Complex Variable Question. Need your correct explanation and answer
ASAP. Thank you!
9. extra points bers a1, a2, Let f(x) be an analytic function on C. Assume there exist complex mum- am not all zero, and a real number q1, such that 72 a)-0 k=1 for all z e C. Show that f() must be a polynomial in the variable z.
9. extra points bers a1, a2, Let f(x) be an analytic function on C. Assume there exist complex mum-...
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t<2n. (a) [5 pts] Show that this curve C lies on the surface S defined by z = 2.cy. (b) [20 pts] By using Stokes’ Theorem, evaluate the line integral| vi F. dr where F(x, y, z) = (y2 + cos x)i + (sin y + z2)j + xk