MATH2403/Tut3.5/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2403: Functions of a Complex Variable
Tutorial 3.5
Combine the things weve learnt recently we can conclude the following
Abels theorem:
Theorem(Abels Theorem). For every p
Math 187
Series problems
1. Find the sum of the geometric series
(a) 3 +
(b)
2
3
3
5
4
9
+
+
3
25
8
27
+
3
125
+
2. Find the rst two nonzero terms of the Maclaurin series of tan x.
3. Find the rst four nonzero terms of the Maclaurin series of
(a) x2 ex
Simpsons Rule
Simpsons rule is a numerical method that approximates the value of a denite integral by using quadratic
polynomials.
Lets rst derive a formula for the area under a parabola of equation y = ax2 + bx + c passing through the
three points: (h, y
MATH2403/LN/NM/ML/2013-14
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
2403 Functions of a Complex Variable
Determination of Coecients in
Partial Fraction Decompositions
Let P (z) = cn z n + cn1 z n1 + + c1 z + c0 ; n = 0; be a polynomial of degr
MATH2403/AS1/Sol/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2403: Functions of a Complex Variable
Solution to Assignment 1
1. (a)
x+i(y1)
x+i(y+1)
=
(x+i(y1)(xi(y+1)
x2 +(y+1)2
x2 +y 2 1
x2 +(y+1)2
=
2x
i x2 +(y+1)2
(b) z 2 z 2 =
MATH2403/NMM/ML/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Summary on Path Integration 2
6. Cauchy Integral Formula
Let f be an analytic function on a disk D(a; R) and r be such other 0 < r < R. The
Cauchy integral formula gives a way
MATH2403/Sol3/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2403: Functions of a Complex Variable
Solution to Assignment 3
1. Proof: we have proved that for f is analytic on D and z D, z+h D, f (z+h)f (z) =
(h)
= 0. For g is analytic
MATH2403/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2403: Functions of a Complex Variable
Tutorial 2 1
2
1. Hyperbolic Function
z
z
We dene the so-called hyperbolic functions. sinhz := e e , coshz :=
2
z
z
z
z
ez +ez
, tanhz := si
MATH2403/Sol-Extra/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2403: Functions of a Complex Variable
Solution to Some Extra Exercises
1. (a) lim (z 1)3 f (z) = 0 a3 = a4 = = 0, lim (z 1)2 f (z) = 4 a2 = 4,
z1
z1
a2
lim (z 1) f (z)
Solving Recurrence Relations
Gilles Cazelais
We want to solve the recurrence relation
an = Aan1 + Ban2
where A and B are real numbers. The solutions depend on the nature of the roots of the characterstic
equation
s2 As B = 0
(1)
We consider three cases fo