# Analytic Function

A single valued function

*f*(*z*) which has a unique derivative w.r.t.*z*at all points of the domain D is said to be analytic at that region.Canuchy-Riemann equation,

=

=

# Cauchyâ€™s Theorem

If

*f*(*z*) is an analytic function and*f*â€²(*z*) is continuous at each point within or on a closed curve C, then = 0If an analytic function

*f*(*z*) within and on a closed curve and if â€˜*a*â€™ is any point within C, then*f*(

*a*) =

# Taylorâ€™s Series

If an analytic function

*f*(*z*) inside a circle C with centre â€˜*a*â€™ then for*z*inside C.*f*(

*z*) =

*f*(

*a*) +

*f*â€²(

*a*) (

*z â€“ a*)

^{2}+ ...... + + ......

**In general :**

# Laurent Series

If an analytic function

*f*(*z*) in a ring shaped region R bounded by two concentric circles C_{1}and C_{2}of radii*r*_{1}and*r*_{2}such that*r*_{1}>*r*_{2}and with centre â€˜*a*â€™, then for all*z*in R.*f*(

*z*) =

*a*

_{0}+

*a*

_{1}(

*z â€“ a*) +

*a*

_{2}(

*z â€“ a*)

^{2}+ ...... +

*a*

_{â€“1}(

*z*â€“ 1)

^{â€“1}+

*a*

_{â€“2}(

*z â€“ a*)

^{â€“2}+ ......

where,

*a*

*=*

_{n}*r*any curve in R encircling C

_{2}.