Cauchy Integral Formula
If 𝑓(𝑧) is an analytic and single valued
within and on a closed contour C and 𝑧0 is
any point within C, then we have
𝑓 (𝑧 )
∮ 𝑑𝑧 = 2𝜋𝑖𝑓 (𝑧0 )
𝐶 𝑧 − 𝑧0
1 𝑓(𝑧)
(or) 𝑓(𝑧0 ) = ∮ 𝑑𝑧
2𝜋𝑖 𝐶 𝑧 − 𝑧0
This expression is known as Cauchy
integral formula. Here, the integral is taken in
the counterclockwise sense.
Derivatives of an Analytic Function
Property:
When 𝒇(𝒛) is an analytic function in a domain
D, then its derivatives of all orders exist and
they also are analytic functions in D.
Derivatives:
The values of derivatives at any point 𝑧0 in the
domain D are as the following:
(𝑛)
𝑛! 𝑓(𝑧)
𝑓 (𝑧0 ) = ∮ 𝑛+1
𝑑𝑧
2𝜋𝑖 𝐶 (𝑧 − 𝑧0 )
′
1! 𝑓(𝑧)
𝑓 (𝑧0 ) = ∮ 𝑑𝑧
2𝜋𝑖 𝐶 (𝑧 − 𝑧0 )2
″
2! 𝑓(𝑧)
𝑓 (𝑧0 ) = ∮ 𝑑𝑧
2𝜋𝑖 𝐶 (𝑧 − 𝑧0 )3
and in general, the 𝑛𝑡ℎ derivative is
(𝑛)
𝑛! 𝑓(𝑧)
𝑓 (𝑧0 ) = ∮ 𝑑𝑧
2𝜋𝑖 𝐶 (𝑧 − 𝑧0 )𝑛+1
where C is a closed contour traversed in the
counterclockwise sense in domain D
surrounding the point 𝑧 = 𝑧0 .
Taylor Series
When 𝑓(𝑧) is analytic at all points inside a
circular domain 𝐷 with its center at 𝑧 = 𝑧0 and
radius 𝑟0 , then for every 𝑧 inside 𝐷, we have
𝑓 ′ (𝑧0 ) 𝑓 ″ (𝑧0 )
𝑓(𝑧) = 𝑓(𝑧0 ) + (𝑧 − 𝑧0 ) + (𝑧 − 𝑧0 )2
1! 2!
𝑓 ‴ (𝑧0 )
+ (𝑧 − 𝑧0 )3 + ⋯
3!
(or)
∞
𝑓 𝑛 (𝑧0 )
𝑓(𝑧) = ∑ (𝑧 − 𝑧0 )𝑛
𝑛!
𝑛=0
This series is known as the Taylor series of
𝑓(𝑧) with center at 𝑧0 .
Special case:
when 𝑧0 = 0, the Taylor series is called the
Maclaurin series of 𝑓(𝑧).
(𝑧 − 𝑧0 )𝑛 becomes (𝑧 − 0)𝑛 = 𝑧 𝑛 .
Maclaurin Series
When 𝑧0 = 0, Taylor series reduces to
Maclaurin Series:
∞
𝑓 𝑛 (0) 𝑛
𝑓(𝑧) = ∑ 𝑧
𝑛!
𝑛=0
Q1) Taylor Series for 𝑓(𝑧) = 𝑒 𝑧 centered
at 𝑧0 = 0
Q2) Taylor Series for 𝑓(𝑧) = sin𝑧 centered at
𝑧0 = 0
Limitation of Taylor Series:
“In many cases it is necessary to expand a
function 𝑓(𝑧) around points where 𝑓(𝑧) is
singular, so that Taylor’s theorem is
inapplicable.”
A singularity is a point where a function
ceases to be analytic. At such points, that
contains a singularity, the derivatives 𝑓 (𝑛) (𝑧0 )
may not exist or be well-defined, so, the
Taylor series expansion invalid.
Note: Taylor series are only applicable in
regions where the function is entirely analytic.
The Necessity of Laurent Series:
To overcome this limitation, the Laurent series
provides a more general type of series
representation.
A Laurent series allows for the expansion
of a function 𝑓(𝑧) around a point 𝑧0 even if 𝒛𝟎
is a singularity,
“For example, if 𝑓(𝑧) is analytic in an annulus
bounded by two concentric circles 𝐶1 and 𝐶2
and at each point on 𝐶1 and 𝐶2 , then Taylor’s
theorem cannot be applied and hence a new
type of series representation of 𝑓(𝑧), that is
Laurent Series is necessary.”
The Laurent series expansion is given by:
∞
𝑓(𝑧) = ∑ 𝑎𝑛 (𝑧 − 𝑧0 )𝑛
𝑛=−∞
∞ ∞
= ∑ 𝑎𝑛 (𝑧 − 𝑧0 )𝑛 + ∑ 𝑏𝑛 (𝑧 − 𝑧0 )−𝑛
𝑛=0 𝑛=1
where
➢The first sum, ∑∞𝑛=0 𝑎𝑛 (𝑧 − 𝑧0 ) , is called the
𝑛
analytic part (or regular part). It contains
non-negative powers of (𝑧 − 𝑧0 ) and is
essentially a Taylor series.
➢The second sum, ∑∞ 𝑏
𝑛=1 𝑛 (𝑧 − 𝑧 0 ) −𝑛
, is
called the principal part. It contains
negative powers of (𝑧 − 𝑧0 ) and it accounts
for the behavior of the function around a
singularity.
In Fig. Two concentric circles 𝑐1 and 𝑐2 of radii
𝑟1 and 𝑟2 respectively, with centre at 𝑧 = 𝑧0 are
connected by cross-cuts.
Let 𝑓(𝑧) be analytic in the annular region
between and on two concentric circles 𝑐1 and
𝑐2 of radii 𝑟1 and 𝑟2 respectively with centre at
𝑧0 .
1 𝑓(𝑧 ′ )𝑑𝑧 ′ 1 𝑓 (𝑧 ′ ) ′
𝑓(𝑧) = ∮ 𝑐1 ′
− ∮ 𝑐2 ′ 𝑑𝑧
2𝜋𝑖 𝑧 −𝑧 2𝜋𝑖 𝑧 −𝑧
Here negative sign is introduced because the
contour 𝐶2 is traversed in the clockwise sense.
Note: Conventionally, a contour in
anticlockwise direction is taken positive and
in clockwise direction is taken negative.
The standard form of a Laurent series is
+∞
𝑓(𝑧) = ∑ 𝐴𝑛 (𝑧 − 𝑧0 )𝑛
𝑛=−∞
1 𝑓(𝑧 ′ )𝑑𝑧 ′
Where, 𝐴𝑛 = ∫
2𝜋𝑖 𝑐 (𝑧 ′ −𝑧0 )𝑛+1
Here,
Z is the point where the function is being
expanded.
𝑧 ′ is the variable of integration
C is a simple closed contour(counter-
clockwise) lying within the annular region
where the Laurent series is valid.
It represents a complex function f(z) as a
sum of terms involving both positive and
negative powers of (𝑧 − 𝑧0 ).
