ƒ: [0, ∞) →R:x→ √ and 8:R¬R:xH Give a formula for the composition g of.

ver the following A) Show that: (r,y)-32y-y-7r is a harmonic function and find the harmonic conjugate function u(x,y).

Evaluate the following complex functions and show your complete solution

The evaluation of f - V4+x2 urz dx is C In(sece + tan0) + c c - In(sec0 + tan0) + c c In|sece – tan0|+ c c -In(sec0 – tan0) + c C None of these MacBo 吕口 F3 F2

Evaluate I = f F. dr, where F(x, y) = (y + sin(x²), x² + e) and C is a circle of radius 5 centered at the origin oriented counterclockwise, using the easiest method. (Use symbolic notation and fractions where needed.) I=

Calculate the Wronskian of functions (3-x*) and 2x. a. W =18x? + 2x° O b. None of these c. W =12x? – 2x° d. W =12x + 2x°

Let f(z) : D C be a complex function. Is it possible for both f(2) and F(z) to be analytic? 2. • If Yes, give an example. • If No, explain why.

a function. tion. (b) Let R= {(x,y) E R × R : 1 < x < 4 or a = 6}, and S = {(x,y) ER × R : –1 < y < 1}. i. Find the relations S1 and R-1 o S. ii. Represent the relations R, S, RUS and RnS in the (r, y)- plane.

Consider the complex-valued function f (x + iy) = x² + ay² – 2xy + i(bx – y² + 2xy). Find values of a and b Correct answers are integers. No spaces, no punctuation except for minus signs if necessary. so that f (x +iy) will be analytic. a= type your answer... and b = type your answer...

Verify directly that the relation y = C, e* cos bx + c, e* sin bx satisfies the equation [(D – a)? + b²]y = 0.