Let ƒ(z) = |z|², z € C. Show that f is complex-differentiable at z = 0, but nowhere else.

Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

Q: . Is there an analytic function f(z) whose real part is u(x, y) = x + e*cos(y) ? If yes, find the…

A: We will find out the required solution.

Q: analytic function a domain D. If a zeck. y) + b v(x, y) = C in D are real Constants not all o. #2.…

A: If f(z)=u+iv is analytic in a domain D , then f satiesfies the Cauchy Riemann (C-R) equations that…

Q: Find and sketch the domain of z = f(x, y) = In(9 - x - y²).

A:

Q: Assume that f(r, y) is a function given by if (r, y) # (0,0), f(r, y) = 0 if (r, y) = (0, 0).

A:

Q: Let f(2)= (2+1)|z|². (2.1) Use the Cauchy-Riemann equations to determine where f is differentiable.…

A: The given function fz=z+1z2 We have to find the points for which fz is differentiable and analytic.

Q: Find the analytic function f (z)u+iv if u = a (1+cose).

A: We have to find the analytic function

Q: Let u > 0 be harmonic in R². Show that u is a constant.

A:

Q: 1. Decide if there is an analytic function f (x + iy) whose real part satisfies Re(f) = xe".|

A: Given problem: To decide analytic function fx+iy whose real part satisfies Ref=xey.

Q: Let f(2) = z2 + 2|2|2 + z where f is differentiable. Determine the set of all points zo where f is…

A:

Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

A:

Q: Let z = x+iy and f(2)=√√|xy|. Show that ƒ(z) satisfies the Cauchy-Riemann equations at the origin,…

A:

Q: Show that the function f(z) = z is nowhere analytic on C.

A:

Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: Let f(2)=u+iv be an analytic function. Given that u=v² show that f(z) is a constant.

A:

Q: Assume f and g are analytic functions in a domain D. If f'(z) = g'(z) for a ll z in domain D, then…

A:

Q: Construct an analytic function of the form f (z) = u + iv, where v is tan- (y/x), x # 0, y # 0

A: f(z)=u+iv where v=tan-1(yx)  We need to find the analytic function f(z)=u+iv  we apply Milne Thomson…

Q: #2. Let flz) = (x, y) +iv(x,y) be an analytic function in a domain D. If a zeck. y) + b v(x, y) = C…

A:

Q: Determine whether the function flz) = e² is analytic or not using the Cauchy- Riemann theorem.

A:

Q: 1) Are the following functions analytic? a) f(z) = z z b) f(z) = z + z?

A:

Q: Let F = (4xyz + 7 sinx, 2x², 2x²y). Find a function f so that F = Vf, and f(0,0,0) = 0.

A: To find the function f such that F=∇f, we need to find the potential function whose gradient is…

Q: y sin x is an i E+ iy is a comp со

A:

Q: Consider a function f given by А f(x, y) u(x) + iv(y). Determine all possible functions f of the…

A: