Module 4C

Application of Complex integration by residues

4.1Evaluation of definite integrals

Where is a rational function of and

Evaluate.

Ans.

Role of the integral are given by

There is only one pole inside the unit circle c

Hence by Cauchy Residue theorem

[sum of the residue within contour]

4.2 Improper Integral

Evaluate.

Ans. Given,

Consider the integral where taken round the close contour c consisting of the upper half of a large circle and the real axis from –R to R.

Poles of are given by

The only pole which lies within the contour is at

The residue of

Hence by Cauchy Residue theorem

Sum of residue

Equating the real part we get

Evaluate 

Answer Consider

Where c is the closed contour consisting of

1)     Real axis from

2)     Large semicircle in the upper half plane given by |z|=R

3)     The real axis -R to and

4)     Small semicircle given by |z|=

Now f(z) has simple poles at z=0 of which only z=is avoided by indentation

Hence by Cauchy’s Residue theorem

Since and

Hence by Jordan’s Lemma

Also since

Hence

Hence as

Equating imaginary parts we get

Prove that

Solution Consider

Where c is the contour consisting of a large semicircle in the upper half plane indented at the origin as shown in the figure

Here we have avoided the branch point o, of by indenting the origin

Then only simple of f(z) within c is at z=i

The residue (at z=i) =

Hence by residue theorem

Since on -ve real axis.

Now

Similarly 

Hence when

Equating real parts we get

TEXTBOOKS/REFERENCES:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons, 2006.
2. J. W. Brown And R. V. Churchill, Complex Variables And Applications, 7th Ed., Mc- Graw Hill, 2004.
3. Veerarajan T., Engineering Mathematics For First Year, Tata Mcgraw-Hill, New Delhi, 2008.
4. N.P. Bali And Manish Goyal, A Text Book Of Engineering Mathematics, Laxmi Publications, Reprint, 2010.
5. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 35th Edition, 2000.