University : University of Madras
Degree : B.Sc
Department :Mathematics
Subject :PAG Complex Analysis
Document type : Question Paper
Website : ideunom.ac.in
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Madras University PAG Complex Analysis Question Paper
U/ID 4707/PAG
Time : Three hours
Maximum : 100 marks
Related : University of Madras PAS Vector Analysis & Analytical Solid B.Sc Question Paper : www.pdfquestion.in/6859.html
OCTOBER 2011
SECTION A — (10 × 2 = 20 marks)
Answer ALL questions.
Each question carries 2 marks.
1. Evaluate 1 1 lim 6 2 + + ® z
2. Using rules of differentiation, find the derivative of cos (2 3 ) 2 z
3. Find the image of a circle throughorigin underthe transformation
4. Define a conformal mapping.
5. Evaluate ? where C is the circle z =1.
6. State maximum modulus principle.
7. Find the zeros of f (z) = sin z . f (z) = sin z
8. Define an isolated singularity for f (z) .
9. State Jordan’s lemma.
10. Find the residue of f(z) = 1-e/z4
SECTION B — (5 × 16 = 80 marks)
Answer ALL questions.
Each question carries 16 marks.
11. (a) State and prove the chain rule for differentiating composite functions.
(b) Derive Cauchy-Riemann equations in Cartesian form.
Or
(c) If f (z) = u + iv and u v e (cos y sin y) – = x – , find f (z) in terms of z .
(d) Show that u(x, y) = sinh x sin y is harmonic and determine its harmonic conjugate. (C) f (z) = u + iv ©ØÖ® u v e (cos y sin y) – = x –
12. (a) If f (z) is analytic in a region D and if f ¢(z) ¹ 0 in D , then prove that the mapping
w = f (z) is conformal in D .
(b) Discuss the transformation w = z .
Or
(c) Find the linear fractional transformation that maps the points z = -i z = and
z = i 3 into the points w = w = i 1 2 1, and 1 3 w = 1.
13. (a) State and prove the Cauchy-Goursat theorem.
Or
(b) State and prove Morera’s theorem.
(c) Expand in a Laurent’s series if z > 3 .
14. (a) Determine and classify the singular points of f(z)=z/e-z
(b) State and prove Cauchy’s residue theorem.
Or
(c) Find the residue of theorem.
(d) State and prove Rouche’s theorem.
15. (a) Evaluate ? – 5 4cos2 cos 3 d .
OCTOBER 2012
U/ID 4707/PAG
Time : Three hours
Maximum : 100 marks
SECTION A — (10 × 2 = 20 marks)
Answer ALL questions.
Each question carries 2 marks.
1. If ( 2 )5 w= 2z +i , then find
2. Write the Cauchy-Riemann equations for the
3. Define Mobius transformation.
4. Write down the linear fractional transformation
5. Define an entire function.
7. Give an example of a function which has aremovable singularity.
8. Find out the zeros of ( ) 2
9. Find the residue of 2 2 1 z + a at z = ai .
10. Define the residue of f (z) at infinity.
SECTION B — (5 × 16 = 80 marks)
Answer ALL questions.
Each question carries 16 marks.
11. (a) Derive the Cauchy Riemann equations in Cartesian form.
(b) Prove that f (z)= z is nowhere analytic.
Or
(c) Show that the function ( 2 2 ) log
(d) Find the analytic function f (z) =u +iv where
12. (a) Prove that the bilinear transformation cz d az b
(b) Discuss the transformation 2 w= z .
Or
(c) Find the bilinear transformation which transforms
(d) Discuss the transformation z w = e .
13. (a) State and prove the Cauchy’s integral formula.
(b) Find the Taylor’s expansion for the function
Or
(c) State and prove the Cauchy’s inequality.
(d) Obtain Taylor’s series for ( ) ( )( ) 1 2
MAY 2011
U/ID 4707/PAG
Time : Three hours
Maximum : 100 marks
SECTION A : (10 × 2 = 20 marks)
Answer ALL questions.
Each question carries 2 marks :
1. If P(z) is a polynomial, find P(z) z z0 lim ®
2. Write Cauchy-Riemann equations in polar form.
3. Define a conformal mapping.
4. Define a bilinear transformation.
5. Evaluate the integral ∫C z dz where C is the upper half of the circle z = 1 .
6. State Cauchy’s inequality.
7. Find a zero of the function f (z) = sin z .
8. State Jordan’s inequality.
SECTION B : (5 × 16 = 80 marks)
Answer ALL questions.
Each question carries 16 marks.
11. (a) Examine whether the function ( ) 2 f z = z is differentiable or not.
Or
(c) Derive Cauchy-Riemann conditions in Cartesian from.
(d) Prove that the function u e y x = cos is harmonic and find a harmonic conjugate.
12. (a) Prove that under a bilinear transformation circles and lines are transformed into circles and lines.
(b) Discuss the transformation w = sin z .
Or
(c) Show that a conformal mapping maps orthogonal curves onto orthogonal curves.