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Questions on Complex Analysis, Exams of Mathematics

Questions on Complex Analysis, Exeecise for foundation on complex numbers and operations

Typology: Exams

2016/2017

Uploaded on 03/13/2017

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Tutorial-II(Complex Analysis)
MS 103: Mathematics-II
Part-A
1. Using definition of differentiability, show that
(a) the function f(z) = Re(z) is nowhere differentiable.
(b) the function f(z) = 1
zis differentiable everywhere except z= 0.
2. Show that the function f(z) = ((z)2
z, z 6= 0;
0, z = 0. is not differentiable at z= 0 but the Cauchy-
Riemann equations are satisfied at (0,0).
3. Obtain the polar form of Cauchy-Riemann equations. Further, if f(z) = u+ivis differentiable
at z=reiθthen show that f0(z) = eiθ(ur+ivr).
4. Find the points at which the following functions are differentiable and evaluate the derivative at
those points:
(a) f(z) = 2(xy +x) + i(x22yy2)
(b) f(z) = x3+i(1 y)3
5. Find the constants aand bsuch that f(z) = (2xy) + i(ax +by) is differentiable for all z.
6. Show that the following functions are entire:
(a) f(z) = eyeix
(b) f(z)=(z22)exeiy
7. Show that the following functions are nowhere analytic:
(a) f(z) = xy +iy
(b) f(z) = eyeix
8. Suppose f(z) is analytic in a domain D. Prove that f(z) must be constant if f(z) is real valued
for all zin D.
9. Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when
(a) u(x, y)=2xx3+ 3xy2
(b) u(x, y) = eycos x
10. (a) If vand Vare harmonic conjugate of uin a domain D, then show that v(x, y) and V(x, y)
can differ by an arbitrary additive constant.
(b) If vis harmonic conjugate of uin a domain Dand uis harmonic conjugate of vthen show
that u(x, y) and v(x, y) must be constant throughout D.
(c) Show in two ways that the function ln(x2+y2) is harmonic in every domain that does not
contain the origin.
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Tutorial-II(Complex Analysis)

MS 103: Mathematics-II

Part-A

  1. Using definition of differentiability, show that (a) the function f (z) = Re(z) is nowhere differentiable. (b) the function f (z) =^1 z is differentiable everywhere except z = 0.
  2. Show that the function f (z) =

{ (^) (z) 2 0 ,^ z ,^ zz^6 = 0;= 0. is not differentiable at^ z^ = 0 but the Cauchy- Riemann equations are satisfied at (0, 0).

  1. Obtain the polar form of Cauchy-Riemann equations. Further, if f (z) = u + iv is differentiable at z = reiθ^ then show that f ′(z) = e−iθ(ur + ivr).
  2. Find the points at which the following functions are differentiable and evaluate the derivative at those points: (a) f (z) = −2(xy + x) + i(x^2 − 2 y − y^2 ) (b) f (z) = x^3 + i(1 − y)^3
  3. Find the constants a and b such that f (z) = (2x − y) + i(ax + by) is differentiable for all z.
  4. Show that the following functions are entire: (a) f (z) = e−yeix (b) f (z) = (z^2 − 2)e−xe−iy
  5. Show that the following functions are nowhere analytic: (a) f (z) = xy + iy (b) f (z) = eyeix
  6. Suppose f (z) is analytic in a domain D. Prove that f (z) must be constant if f (z) is real valued for all z in D.
  7. Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x − x^3 + 3xy^2 (b) u(x, y) = ey^ cos x
  8. (a) If v and V are harmonic conjugate of u in a domain D, then show that v(x, y) and V (x, y) can differ by an arbitrary additive constant. (b) If v is harmonic conjugate of u in a domain D and u is harmonic conjugate of v then show that u(x, y) and v(x, y) must be constant throughout D. (c) Show in two ways that the function ln(x^2 + y^2 ) is harmonic in every domain that does not contain the origin. 1

Part-B

  1. Show that (a) e^2 ±^3 πi^ = −e^2 (b) e 2+^4 π^ i= √^ e 2 (1 + i)
  2. Find all values of z such that (a) ez^ = − 4 (b) e^2 z−^1 = 1
  3. Show that |ez^2 | ≤ e|z|^2 for all z. When does equality occur?
  4. Show that (a) ez^ = ez^ for all z. (b) eiz^ = eiz^ if and only if z = nπ, n = 0, ± 1 , ± 2 , · · ·.
  5. Show that if ez^ is real, then Im z = nπ, n = 0, ± 1 , ± 2 , · · ·.
  6. Show that |e−z^ | < 1 if and only if Re(z) > 0.
  7. Prove that the function ez^ is not analytic anywhere.
  8. Find all values of (a) Log(ie^2 ) (b) Log(1 + i)^4 (c) log(4i) (d) log(−2 + 2i)
  9. Show that Log(1 + i)^2 = 2Log(1 + i) but that Log(−1 + i)^2 = 2Log(−1 + i).
  10. Find all roots of the equation log z = 1 − ( π 4 )i.
  11. Show that (a) the function Log(z − i) is analytic everywhere except on the half line y = 1 (x ≤ 0). (b) the function Log( z (^2) +3zz+5)+2 is analytic everywhere except at the points −1, −2 and on the portion x ≤ −5 on the real axis.