Indian Institute of Technology Guwahati
Department of Mathematics
MA 201 Mathematics-III
July-November 2013
Tutorial Sheet - 7
1. Use Taylors series expansion of eiz and eiz around 0 to get the Taylors series expansion of cos z around 0.
2. Let C be a sim
MA 201
COMPLEX ANALYSIS
ASSIGNMENT4
1
(1) Is there a polynomial P (z) such that P (z)e z is an entire function? Justify
your answer.
1
Answer: No. e z has innitely many terms in the principal part of the Laurent
1
series about 0. If we multiply any polyno
MA 201
COMPLEX ANALYSIS
ASSIGNMENT1
Notation: D = cfw_z C : |z| < 1.
(1) Prove the following:
(a) Prove that |z| |Re (z)| + |Im(z)| 2 |z|.
Answer: Observe that, for any two real numbers x and y, we have
(|x| |y|)2 0 = |x|2 + |y|2 2|x| |y|.
Let z = x + iy
MA 201
COMPLEX ANALYSIS
ASSIGNMENT5
(1) Evaluate
Answer:
|z |=
2
2
z dz
.
cos z
(2n + 1)
for n Z, Therefore, the function
2
z
(2n + 1)
f (z) =
has a simple poles at
for n Z. Inside the contour
cos z
2
the function f has only one pole z = 2 which is a sim
Open Set and Closed set
Lecture 2
Open Set & Closed set
Some Basic Denitions
Open disc: Let z0 C and r > 0 then, B(z0 , r ) = cfw_z C : |z z0 | < r
is an open disc centered at z0 with radius r .
Deleted Neighborhood of z0 : Let z0 C and r > 0 then,
B(z0
Sequence, Limit and Continuity
Lecture 3
Sequence, Limit and Continuity
Functions of a complex variable
Let S C. A complex valued function is a rule that assigns to each
complex number z S a unique complex number w .
We write w = f (z). The set S is calle
Analytic functions
Lecture 5
Analytic functions
Analytic functions
Denition: A function f is called analytic at a point z0 C if there exist
r > 0 such that f is dierentiable at every point z B(z0 , r ).
A function is called analytic in an open set U C if
Dierentiability
Lecture 4
Dierentiability
Dierentiability
Recall: Let A be a nonempty open subset of R. x0 A. Then we say f is
dierentiable at x0 if the limit
lim
h0
f (x0 + h) f (x0 )
h
exists.
Denition: Let D be a nonempty open subset of C. z0 D. Then f
Elementary functions
Lecture 6
Elementary functions
The Exponential Function
Recall:
Eulers Formula: For y R, e iy = cos y + i sin y
and for any x, y R, e x+y = e x e y .
Denition: If z = x + iy , then e z or exp(z) is dened by the formula
e z = e (x+iy )
Elementary properties of Complex numbers
Lecture 1
Complex Analysis
Introduction
Let us consider the quadratic equation x 2 + 1 = 0.
It has no real root.
Let i(iota) be the solution of the above equation, then
i 2 = 1 i.e. i = 1.
i is not a real number. S
Complex Integration
Lecture 7
Complex Integration
Complex Integration
Integral of a complex valued function of real variable:
Denition: Let f : [a, b] C be a function. Then f (t) = u(t) + iv (t)
where u, v : [a, b] R.Dene,
b
b
f (t)dt =
a
b
u(t)dt + i
v (
Complex Integration
Lecture 8
Complex Integration
Recall
Denition: Let (t); t [a, b], be a contour and f be complex valued
continuous function dened on a set containing then the line integral or the
contour integral of f along the curve is dened by
b
f (t
Applications of Cauchys Integral Formula
Lecture 11
Applications of Cauchys Integral Formula
Cauchys estimate
Cauchys estimate: Suppose that f is analytic on a simply connected domain
D and B(z0 , R) D for some R > 0. If |f (z)| M for all z SR (z0 ), then
Cauchy Integral Formula
Lecture 10
Cauchy Integral Formula
Cauchy Integral Formula
Theorem Let f be analytic on a simply connected domain D. Suppose that
z0 D and C is a simple closed curve oriented in the counterclockwise in D
that encloses z0 . Then
f (
MA 201
COMPLEX ANALYSIS
ASSIGNMENT3
(1) Show that
Answer:
eaz
dz =
2
z +1
eaz
dz = 2i sin a, where (t) = 2eit , t [0, 2].
z2 + 1
eaz
1
1
1
1
dz =
2i z i z + i
2i
eaz
dz
zi
eaz
dz
z+i
By Cauchy integral formula,
1
2i
eaz
dz
zi
2
eaz
1
dz = 2i [eia eia ]
MA 201
COMPLEX ANALYSIS
ASSIGNMENT2
(1) Find the values of z such that (a) ez R and (b) ez iR.
Answer:
ez = ex+ i y = ex (cos(y) + i sin(y) .
We know that ex = 0 for all x R.
ez is a real number i sin y = 0 i y = n where n Z.
ez is a pure imaginary number
MA 201
COMPLEX ANALYSIS
ASSIGNMENT2
(1) Find the values of z such that (a) ez R and (b) ez iR.
(2) Prove that sinh(Imz) | sin(z)| cosh(Imz). Deduce that | sin(z)| tends to
as |Imz| .
(3) Find all the complex numbers which satisfy the following:
(i) exp(z
MA 201 (Part II)
Partial Differential Equations
Session July-Nov, 2014
Solutions to Tutorial Problems - 1
Topics: Derivation of PDEs, General integrals, Integral surface, Cauchy problem
1. Find the partial dierential equation arising from each of the foll
MA 201 (Part II)
Partial Differential Equations
Tutorial Problems - 8
Separation of variables method: one-dimensional wave equation and heat conduction equation
and Laplaces equation
1. A string is xed at its ends x = 0 and x = l and lies initially along
MA 201 (Part II)
Partial Differential Equations
Session July-Nov, 2014
Solutions to Tutorial Problems - 7
Topics:Classication of 2nd order PDEs, Canonical/Normal forms
The wave equation:Innite string problems, semi-innite string problems, nite vibrating s
MA 201
COMPLEX ANALYSIS
ASSIGNMENT3
(1) Show that
2
eaz
dz = 2i sin a, where (t) = 2eit , t [0, 2].
2+1
z
i
ee d.
(2) Evaluate
0
f (z)
= 0. Show that f is constant.
z
(4) Let f : C C be a function which is analytic on C \ cfw_0 and bounded on
B(0, 1 ). Sh
MA 201
COMPLEX ANALYSIS
ASSIGNMENT1
Notation: D = cfw_z C : |z| < 1.
(1) Prove the following:
(a) Prove that |z| |Re (z)| + |Im(z)| 2 |z|.
(b) |z1 + z2 | |z1 | + |z2 | and equality holds if and only if one is a nonnegative
scalar multiple of the other.
z1
MA 201 (Part II)
Partial Differential Equations
Tutorial Problems - 6
1. Find the partial dierential equation arising from each of the following surfaces:
(a) z = f (x y), (b) 2z = (ax + y)2 + b, (c) log z = a log x + 1 a2 log y + b,
(d) f (x2 + y 2 , x2
MA 201
COMPLEX ANALYSIS
ASSIGNMENT4
(1) Evaluate
|z |=
2
2
z
dz.
cos z
(2) Using the Cauchys residue theorem, evaluate
|z|=2
(3)
(4)
(5)
(6)
(7)
(8)
(z 2 + 3z + 2)
dz.
(z 3 z 2 )
1
Using the argument principle, evaluate
cot z dz where C : |z| = 7.
2i C
L
MA 201
COMPLEX ANALYSIS
ASSIGNMENT4
1
(1) Is there a polynomial P (z) such that P (z)e z is an entire function? Justify
your answer.
1
(2) Find the Laurent series of the function f (z) = exp z + z around 0. Hence
show that for all n 0
1
2
2
e
2 cos
cos n
MA 201 (Part II)
Partial Differential Equations
Tutorial Problems - 8
Separation of variables method: one-dimensional wave equation and heat conduction equation
and Laplaces equation
1. A string is xed at its ends x = 0 and x = l and lies initially along
MA 201 (Part II)
Partial Differential Equations
Tutorial Problems - 9
BVP/IBVP in polar and cylindrical coordinates
Fourier integral and Fourier transform
1. Consider transient heat conduction in a circular region of radius a. Considering that heat
conduc
MA 201 (Part II)
Partial Differential Equations
Tutorial Problems - 10
Laplace Transform: Properties and applications
1. Find the Laplace transforms of
t
3t
(i) te cos 4t, (ii) t
e
3t
sin 2t dt, (iii)
0
0
t
e3t sin 2t
dt.
t
2. Find the Laplace transform o
Cauchys Theorem
Lecture 1
Cauchys Theorem
Complex integration
Question: Under what conditions on f we can guarantee the existence of F
such that F = f ?
Denition:(Simply connected domain) A domain D is called simply
connected if every simple closed contou