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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,
Description: 397.png= Description: 402.png 
Description: 407.png= Description: 412.png 

Cauchy’s Theorem

If f (z) is an analytic function and f (z) is continuous at each point within or on a closed curve C, then Description: 417.png= 0
If an analytic function f (z) within and on a closed curve and if ‘a’ is any point within C, then
f (a) = Description: 422.png

Taylor’s Series

If an analytic function f (z) inside a circle C with centre ‘a’ then for z inside C.
f (z) = f (a) + f (a) (z – a)2 + ...... + Description: 432.png + ......Description: 427.png
In general : Description: 437.png 

Laurent Series

If an analytic function f (z) in a ring shaped region R bounded by two concentric circles C1 and C2 of radii r1 and r2 such that r1 > r2 and with centre ‘a’, then for all z in R.
f (z) = a0 + a1 (z – a) + a2 (z – a)2 + ...... + a–1 (z – 1)–1 + a–2 (z – a)–2 + ......
an = Description: 442.png
r any curve in R encircling C2.

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