MA8251 Notes Engineering Mathematics 2 Unit 4

MA8251 Notes Engineering Mathematics 2 Unit 4 COMPLEX INTEGRATION Regulation 2017 For Anna University Free download. ENGINEERING MATHEMATICS 2 MA8251 Unit 4 COMPLEX INTEGRATION Notes Pdf Free download.

Content Engineering Mathematics 2 ma8251 Unit 4 COMPLEX INTEGRATION

COMPLEX INTEGRATION

1. Introduction

2. Cauchy’s Theorem

2.1 Definitions

2.1.1Connected Region

A connected region is one which any two points in it can be connected by a curve which lies entirely with in the region.

2.1.2 Simply connected region

A curve which does not cross itself is called a simple closed curve. A region in which every closed curve in it encloses points of the region only is called a simply connected region. (ma8251 notes engineering mathematics 2 unit 4)

2.1.3 Contour integral

An integral along a simple closed curve is called a contour integral.

2.1.4 Cauchy’s Integral Theorem

If a function f(z) is analytic and its derivative f0(z) is continuous at R all points inside and on a simple closed curve c, then c f(z)dz = 0:

2.1.5 Cauchy’s Integral formula

If f(z) is analytic inside and on a closed curve c of a simply connected region R and if a is any point with in c, then (ma8251 notes engineering mathematics 2 unit 4)

2.1.6 Cauchy’s integral formula for derivative

If a function f(z) is analytic within and on a simple closed curve c and a is any point lying in it, then

3. Taylor’s and Laurent’s

3.1 Taylor’s Series.

A function f(z), analytic inside a circle C with center at a, can be expanded in the series

3.2 Laurent’s Series.

Let C1; C2 be two concentric circles jz aj = R1 and jz aj = R2 where R2 < R1: Let f(z) be analytic on C1andC2 and in the annular region R between them. Then, for any point z in R, (ma8251 notes engineering mathematics 2 unit 4)

4. Singularities

5. Residues

6. Evaluation of real definite Integrals as contour integrals

7. Applications (ma8251 notes engineering mathematics 2 unit 4)

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