(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
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 for inf
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 converg
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 ,
MSO202A: Introduction To Complex Analysis
Dr. G. P. Kapoor
FB 565, Department of Mathematics and Statistics
Tel. 7609, Email: [email protected]
E. Kreyszig, Advanced Engineering Mathematics, 8th Ed., John
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
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