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 CDSE

Department of Engineering Mathematics

  Subject Code: BE 401 Subject: Engineering Mathematics –III

utoria! opic

1 Analytic Function, C-R Equations

" Complex Integration

# Poles , Residue, Singularity

4$ Contour Integration

utoria! – 1

%eca!!: 1. Functions o complex !aria"le, its limit, continuity and dierentia"ility

#. Analytic unctions and t$eir properties

%. &ierence "et'een poles, residue and singularity

(. Contour integration

&1$ I),*   y xu

 and),*   y xv

  are $armonic unctions in a region R, pro!e t$at

   

  

 ∂∂

+∂∂

+   

  

 ∂∂

−∂∂

 y

v

 x

ui

 x

v

 y

u

  is an analytic unction o  z = x+ i y .

&$" Find t$e imaginary part o t$e analytic unction '$ose real part is

% x  ### %%%   y x xy   −+−

$ S$o' t$at t$e unction f  ( z )= xy+iy  is continuous e!ery'$ere "ut is not analytic.

&$4 I)* z  f  

 is a regular unction o z 

 pro!e t$at

#+#

#

#

#

#

)*()*   z  f   z  f   y x

=   

  

 ∂∂

+∂∂

&$' E!aluate ∫0

2+i

( ´ z )2 dz ,  along

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*i) $e real axis to # and t$en !ertically to #i.

*ii) $e line y=

1

2 x

&$( Find t$e order o eac$ pole and residues at it or t$e unction1

1( + z 

.

&$):  S$o' t$at an analytic unction 'it$ constant modulus is constant

&$* using Cauc$ys integral ormula, e!aluate ∫c

❑e2  z

( z+1 )4 dz

 , '$ere C is t$e circle I / I 0%.

&$+ E!aluate ∫C 

❑cosπ z

2

( z−1 )( z−2)dz

, '$ere C is t$e circle I/I 0%.

&$10 sing contour integration E!aluate

#

2  # cos

d π θ

+ θ∫ 

&$11 Pro!e t$at

( )11

#

cos#1

#cos   ##

##

2

#  <−

=+−∫ 

  aa

ad 

aa

π θ 

θ 

θ π  

,ns-ers:

3#.   3 x2

 y+6 xy− y3+c , '$ere c is a constant.

3.4 *i)

1

3(14+11i)

  *ii)

5

3(2−i)

3.5−14 (1+i√ 2 ) ,

−14 (−1+i√ 2 ) ,

−14 (−1−i√ 2 ) ,

−1

4 (1−i√ 2 )

3.68π e

−2

3i

3.7 (   πi

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3.12

2 π 

√ 3