CHAPTER 2 – COMPLEX DIFFERENTIATION
2marks :
1. Show that _____ is analytic.
2. Show that _______ is harmonic.
3. Test the analyticity for _____.
4. Check Whether ______ can be real part (or) imaginary part of an
analytic function.
5. Find the Critical points for ________.
6. Find the invariant Points (OR) Fixed points of ______.
7. Find f(z) if real part (OR) imaginary part is given as _____.
8. Find a , b , c if f(z) = ______ to be analytic.
( Note
: “ _____ “ implies that model problem
will come “)
9. Show that analytic function with Constant Real part (OR) Constant
imaginary Part (OR) Constant modulus (Argument) part is constant.
10. State the properties of analytic function.
11. State necessary & sufficient conditions for f(z) to be analytic.
12. Define Analytic (or) Regular (or) Holomorphic function.
13. Write C.R equations in Cartesian form (OR) Polar Form.
14. Define Bilinear (OR) Mobius Transformation.
15. Define Confirmal Mapping.
16. Define Critical point.
17. Define Singular Point.
18. Define Invariant Point (OR) Fixed Point.
19. Define Isogonal Transformation.
20. Define Cross ratio.
21. If u , v are analytic , prove that u+iv is also analytic.
22. If u , v are harmonic , prove that u+iv is also analytic.
23. If u+iv is analytic , prove that v-iu is also analytic.
24.If f(z) is analytic , prove that k x f(z) is also analytic. (where
k is constant).
25. What is the necessary condition for
the existence of the derivative of f(z)?
26. If w = log z, then determine where w
is non analytic.
27. Define singular point of the function
with an example.
28. Show that f(z) = xy + iy is not
analytic.
29. Define holomorphic function.
30. Show that v = ex
sin y is harmonic function.
31. Show that v = ex
cos y is harmonic function.
32. Find the invariant points of the
transformation w = z−1/z+1.
33. Under the transformation w = 1/z
, find the image of |z − 2i| = 2.
34. Show that the transformation w = 1/z transforms all circles and straight
lines to circles and straight lines in the w-plane.
35. State two important properties of
Mobius transformation.
36.Prove that
the real and imaginary part of an analytic function satisfies Laplace
equations.
37. Is f (z ) = z3analytic ?
Justify.
38. Prove that z z is nowhere
analytic.
39 .For what values
of a,b and c the function
f (z) = x -2ay + i(bx - cy) is analytic.
40. If u+iv is
analytic , show that v – iu &-v + iu are also analytic.
41. State the
orthogonal property of an analytical function.
42.
Write down the formula for finding an
analytic function f (z) = u + iv, whenever the real part is given by using
Milne Thomson method.
43. Find
‘a’ so that2–y2 +xyu (x,y)isharmonic=. ax
(Note
: If they Ask , Prove that log z is analytic , at las analytic except at z=0 )
15 marks :
(You can Omit Atmost one Part Mentioned Below)
PART 1 – Find
the analytic function whose u (OR) v (OR) u+v (OR) u-v is given as
PART 2 – Confirmal Mapping.
PART 3 – Bilinear Transformation
PART 4 - Proof Sum.
PART 5 – Image of Transformation Sum .
Part 6 - Problem Using Milne
Thomson Method.
Proof Sums:
3.Prove that “If v is a harmonic conjugate of u, then the two families of curves u(x, y) = α and v(x, y) = βare mutually orthogonal to each other’’.
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