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) = e−iθ(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(x2−2y−y2)
(b) f(z) = x3+i(1 −y)3
5. Find the constants aand bsuch that f(z) = (2x−y) + i(ax +by) is differentiable for all z.
6. Show that the following functions are entire:
(a) f(z) = e−yeix
(b) f(z)=(z2−2)e−xe−iy
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)=2x−x3+ 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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