Complex Variables : Differentiation - Quotient Formula

7. Prove that dzn-1/ dz = nzn-1, for the derivative of zn remains valid when n is a negative integer (n = -1, -2, •••), provided that z ≠ 0. (Suggestion: Write m = -n and use the formula for the derivative of a quotient of two functions.)

Complex Variables : Differentiability

9. Let f denote the function whose values are _ f (z) = z^2 / z when z ≠ 0, f (z) = 0 when z = 0. Show that if z = 0, then ∆w/∆z = 1 at each nonzero point on the real and imaginary axes in the ∆z, or ∆x∆y, plane. Then show that ∆w/ͧ ...continues

Abel’s Theorem: ∞ If the power series ∑ an zn converges for a particular value zo of z, then it converges absolutely n = 0 for every z for which │z│< │zo│.

Complex Variables Power Series (I) Abel’s Theorem: ∞ If the power series ∑ an zn converg ...continues

Use the given theorem to show that each of these functions is differentiable in the indicated domain of definition, and then find f ΄(z):

The problems are from complex variable class. Please specify the terms that you use if necessary and explain each step of your solution. If there is anything unclear in the problem, please tell me. Thank you very much. 4. Use the given theorem to show that each of these functions is differentiable in the indicated domain of ...continues

Cauchy-Riemann Equations

6. Let u and v denote the real and imaginary components of the function f defined by the equations _ f(z) = (z^2)/z when z ≠ 0, f (z) = 0 when z = 0. Verify that the Cauchy-Riemann equations ux = vy and uy = -vx are satisfied at the origin z = (0, 0).

Cauchy-Riemann Equations

Please see the attached file for the fully formatted problems. 7. Solve equations (2) for ux and uy to show that ux = ur cos θ – (uθ sin θ) / r , uy = ur sin θ + (uθ cos θ) / r. Then use these equations and similar ones for vx and vy to show that equations (4) are satisfied at a point z0 ...continues

Cauchy-Riemann Equations

Please see the attached file for the fully formatted problems. 8. Let a function f (z) = u + i v be differentiable at a nonzero point z0 = r0 e(iθ0). Use the expressions for ux and vx found in Exercise 7, together with the polar form (6) of Cauchy-Riemann equations, to rewrite the expression f ΄(z0) = ux + i vx ...continues

Entire Functions

1. Apply the given theorem to verify that each of these functions is entire: (a) f (z) = 3x + y + i (3y - x) (b) f (z) = sin x cosh y + i cos x sinh y (c) f (z) = e-y sin x - i e-y cos x (d) f (z) = (z2 - 2) e-x e-iy.

Determine the singular points of the function

4. In each case, determine the singular points of the function and state why the function is analytic everywhere except at those points:... see attachment

Show that the composite function G (z) = g (2z – 2 + i) is analytic in the half plane

Show that the composite function G (z) = g (2z – 2 + i) is analytic in the half plane x > 1, with derivative .... see attachment