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PYL113 - Mathematical Physics
Tutorial Sheet 1(A) - Complex Analysis
1. Find the equation for a circle in the complex plane, which
passes through threepoints z1, z2 and z3.
2. Study the differentiability ofzez. Is it an entire
function?
3. Determine the type of singularities possessed by the
following functions:(a) (z2)1 (b) (1+z3)/z2 (c)ez/z3 (d) (z
+3i)5/(z22z +5)(e) sec(1/z) (f) sin(
z)/
z
4. Using f(rei) = R(r, )ei(r,), in which R(r, ) and (r, ) are
differentiable realfunctions of r and, show that the Cauchy-Riemann
conditions in polar coordinates
read Rr
= R
r
, 1
rR
= R r
5. Two dimensional irrotational fluid flow is conveniently
described by a complex po-tentialf(z) =u(x, y)+iv(x, y). We label
the real part,u(x, y), thevelocity potential
and the imaginary part, v(x, y), the stream function. The fluid
velocity V is given
by V = u. Iff(z) is analytic, prove that(a) df/dz= Vx+iVy.
(b) V= 0 (no sources or sinks).(c) V= 0 (irrotational,
nonturbulent flow).
6. For a functionf(z), analytic in a simply-connected region R,
prove that the integralba
f(z)dz, is the same for all paths between a and b, provided the
path, as well asthe end points a, b R.
7. Show that
(a) 1
2i
C
zmn1dz = mn , wheremn is the Kronecker Delta function
(b) 12i
lim
dei
+i=() , where() is the Unit Step function
The above relations constitute integral representationsof the
stated functions.
8. Investigate the convergence properties of the following
series:
(a)n=1
zn/(3n + 1), (b)n=0
e2inz/(n+ 1)3/2
9. Find the radii of convergence of the following Taylor
series:
(a)n=2
zn/ ln n (b)n=1
znnlnn
10. Expand the following functions in an appropriate power
series, and specify theregion(s) of convergence:
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(a) ez2
sinh(z+ 2), about z0= 0.
(b) (1 cos z)/z, about z0= 0.
11. Derive the Laurent series expansion of an analytic function
f(z) (working out allthe intermediate steps.)
12. Prove that iff(z) has a simple zero at z0, then 1/f(z) has a
residue of 1/f(z0).
Hence evaluate the integral:
sin
a sin d
wherea is real and > 1.
13. Use the residue theorem to evaluate:
(a)
C
ez+1/z dz, C being|z| = 1
(b)
C
1
sin3 zdz, C being|z| = 2
(c)
20
1
a+b cos d,
(d)
0
coshax
coshx dx, |a|
