To Find Radius of Convergence From Ratio of Consecutive
Terms an / an 1 of the Power Series
a n ( z z0 )
, an 0
for all n .
The formula for radius of convergence in terms of an 1 / an
does not work if an = 0
A series of the form
an ( z z0 )n
is called a power series. The complex numbers an s are called
the coefficients and z0 is called the centre of the power series.
For what values of z a power series converges?
Line Integrals Independent of Path
Definition (Simply Connected Domain): A domain G is called
simply connected if every simple closed curve in G encloses only
points of G (i.e. the domain G has no holes).
Let G be
Maximum Modulus Theorem: If f is analytic in a domain D and
if there is a point a D such that f ( a ) f ( z ) for all z D ,
then f is a constant function.
The above theorem can also be stated as A nonconstant
We show that every analytic function can be expanded into a
power series, called the Taylor series of the function.
Taylors Theorem: Let f be analytic in a domain D & a D. Then,
f(z) can be expressed as the power series
Let f be analytic inside and on a simple, closed, piecewise
smooth curve C. Then,
f ( z ) dz 0 .
Definitions: Let z (t ), a t b , be parametric representation of
the curve C.
Simple Curve: Th
Behaviour of f ( z ) in the neighbourhood of Pole:
Proposition. The point a is a pole of order m of f iff
lim( z a ) m f ( z ) A, A 0, .
(i) If the point a is a pole of order m of f, then
f ( z
Singularities of a Complex Function
A point a is called a singularity of a function f ( z ) if f ( z ) is
not analytic at the point a .
A singularity a is called an isolated singularity of f ( z ) , if
f ( z ) i
(II) Integrals of the form
f ( x ) dx is defined as
f ( x ) dx .
f ( x ) dx lim
f ( x ) dx lim f ( x ) dx .
If the limit on RHS does not exist, or gives an indeterminate
f ( x ) dx
Definition: Let C be a simple closed p.w. smooth curve enclosing
all finite singularities of f. Orient C in clockwise direction.
Then, residue of f at z is defined as
res f ( z )
f ( z ) dz .
Mapping Properties of Analytic Functions
Example. Consider f ( z ) z 2 , z x iy and w f ( z ) u iv .
u x 2 y 2 , v 2 xy (*)
Using (*), the following mapping properties of f ( z ) z 2 are
Example. Prove that the circles of Apollonius K :
k , k 1, are orthogonal to any circle C passing through
the points p and q .
Consider the Mobius transformation w
. It maps
(i) the circle K oneone onto th
A special class of Conformal Mappings: Mobius
Transformations or Linear Fractional Transformations
A function S(z) of the form S ( z )
, ad bc 0 , is called a
The complex numbers a
Properties of Logarithmic Function (Contd)
Since, Log z ln z i Arg z
u Re Log z ln( x 2 y 2 )
v Im Log z tan 1 constant
vy , uy 2
It follows that u x 2
This shows that Re Log z and Im Log z
MSO202: Introduction To Complex Analysis
) 0, b 0
We first give the geometrical of H 0 cfw_z : Im( ) 0, b 1 , i.e.
when a = 0 and b 1.
In this case, write z r (cos i sin ) and b cos i sin ,
Basic Properties of Differentiable Functions
Proposition 1. Let f : be defined in some
neighbourhood of a and differentiable at a. Then,
( a ),
( a ) exist and satisfy
( a ) i ( a ) . (*)
F E AT U R E
2-Cycle Moment Distribution For The Analysis of
Continuous Beams And Multi-Storey Framed Structures
By: Ir. Chan Kam Leong, B.Sc(Eng.), MSc, P.Eng, C.Eng, MICE, MIStruct E, MIEM, MACEM
K.L. Chan & Associates, Consulting Engineers
Prof. M. S. Sivakumar
Strength of Materials
Deflection of beams
Deflection of Beams (Solution Method by Direct Integration)
Moment - Area Method for finding Beam Deflections
Indian Institute of Technology Madras
Strength of Materials
Deflection: Virtual Work
Method; Beams and
Theory of Structure - I
of Virtual Work:
Method of Virtual Work : Beams and Frames
Virtual unit load
MSO202A: Introduction To Complex Analysis
Dr. G. P. Kapoor
FB 565, Department of Mathematics and Statistics
Tel. 7609, Email: firstname.lastname@example.org
E. Kreyszig, Advanced Engineering Mathematics, 8th Ed., John