Unit- IV
ANALYTIC FUNCTIONS
PART-A
𝑥
1. Show that 𝑥 2 +𝑦2 is harmonic.
2. Is f(z) = z3 analytic?
1
3. Find the invariant point of the transformation w = 𝑧−2𝑖
4. Show that xy2 cannot be the real part of an analytic function.
5. Find the image of x2+y2 = 4 under the transformation w=3z.
6. f(z) = u + iv is such that u and v are harmonic is f(z) analytic always? Justify.
7. State the Cauchy-Riemann equations in polar coordinates satisfied by an analytic function.
2𝑧+6
8. Find the invariant points of the transformation w= 𝑧+7 .
9. Find the analytic region of f(z) = (x-y)2+2i (x+y).
10. Find the critical points of the transformation w2= (z-𝛼)(z-𝛽).
11. Define conformal mapping.
12. For what values of a,b and c the function f(z) = x – 2ay + i(bx-cy) is analytic?
13. If u+iv is analytic, show that v-iu is also analytic.
14. Find the image of the circle |z| = 2 under the transformation w = 3z.
15. Define bilinear transformation.
16. Define analytic function of a complex variable.
17. Give an example such that u & v are harmonic but u+iv is not analytic.
18. Find ‘a’ so that u(x,y) = ax2-y2+xy is harmonic.
19. State the orthogonal property of an analytic function.
20. Under the transformation w = iz + I show that the half plane x>0 maps into the half plane w>1.
21. Find the points in the z plane at which the mapping w = z + 𝑧 −1 , (z≠0) fails to be conformed.
𝑦
22. Prove that tan−1 𝑥 is harmonic.
23. Show that the function f(z) = zz̅ is not analytic at z=0.
24. f(z) = r2 (cos 2𝜃 +i sin 𝑝𝜃) is analytic if the value of p is ….?
a) ½
b) 0
c) 2
d) 1
25. Define Mobius transformation.
1
26. If u ≠ iv = 𝑧; then prove that the families of curves u = c1 and v = c2 ( c1 , c2 being constants) cut orthogonally.
27. Define Isogonal transformation.
28. Verify whether w = sin 𝑥 cos ℎ𝑦 + 𝑖 cos 𝑥 sin ℎ𝑦 is analytic or not.
29. Find the bilinear transformation which maps the points z = -2, 0, 2 into the points w = 0, i, -i respectively.
𝜕2
𝜕2
30. If f(z) is a regular function of z, prove that (𝜕𝑥 2 + 𝜕𝑦2 ) |f(z)|2 = 4 |f ’(z)|2