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7. Let f:D + C be a complex variable function, write f(x) = u(x, y) +iv(x,y)...
14. Determine whether or not the complex function f(z) -4x2+5x -4y2 + i(8xy 5y+3) is analytic for all z є c.
Problem 6 (6 points) Let f(x) = u(x, y) + iv(x, y) be a analytic function on D and extends continuously to ad. Prove that the component function u(x, y) must attain its minimum value on aD unless u(x,y) is a constant function. (Hint: Consider the modulus of analytic function g(z) = ef(x), and apply the result in problem 5)
Problem 8. Let f(z) = u(x, y) iv(x, y) be an entire function with real and imaginary parts u(x, y) and v(x, y). Assume that the imaginary part is bounded v(x, y) < M for every z = x+ iy. Prove that f is a constant 1
2- a) The real part of a complex function f(z) given as, u(x, y) = 3x?y - y. Iff(2) is an analytic function, find v(x,y) and f(z) (15p) b) Find the whether f(z) is analytic or not where f(z) = cos(x) +ie'sinx. (15p)
Problem 1 Consider the composition f(w(z)) of two complex valued functions of a complex variable, f(w) and w(z), where z = x+iy and w=u+iv. Assume that both functions have continuous partial derivatives. Show that the chain rule can be written in complex form as of _ of ou , of Oz . . of az " dw dz * dw dz and Z of ou , of ou dw dz* dw ƏZ Show as a consequence that if f(w) is...
Problem A.5. Let D be a region in the complex plane. (a) State Green's theorem in terms of f(2)u(, y) + iv(x, y),z-+ iy, and (b) Prove the following case of Morera's theorem: If f is continuously differentiable 0 for every circle γ in D, then f is analytic in D. Hint: in D and J,f(z)dz Use part (a).
complex anaylsis f(z) = f(x+iy) = (x + 3xy - by ²x) + (y3 + 3x²y+y) Find all points Zec at which differentiable. Then, find all the points at which ZEC t is analytic.
Let u = 2.c" + 2x - 2y + 3y. (a) (6 points) Show that u is harmonic (b) (10 points) Find a harmonic conjugate of u. (C) (9 points) Find an analytic function f(x) of z = x + iy such that Ref = u.
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-...
Let f(x, y) = 5x2 - 4x + 5y + 4y2 + 8xy List the critical point: Preview The critical point is a Select an answer Get help: Video