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ba bsc 6 sem mathematics complex analysis s 253 b 2022

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S-253(B)

B. A./B. Sc. (Sixth Semester)

EXAMINATION, 2021-22

MATHEMATICS

(Complex Analysis)

[SOS/Maths./DSE-002(B)]

Time : Two Hours]                                                                  [Maximum Marks : 70

 

Note: (1) Attempt any five questions from Section A and any three questions from Section B.

(ii) Answer each question of Section A within 50 words.

(iii) Limit your answers within the given answer book. Additional answer book (B-Answer book) should not be provided or used.

                                                                                                         

Section-A

note : Attempt any five questions. Each question carries 5 marks.

1. Show that, if lim f (z) = l and lim g(z) = m then

z®Z0                 z®z0

= provided that m¹ 0.

 

2. Show that f(z) = 1/z is not uniformly continuous in the domain | z |<|.  

 

3. Show that the function f(z) = xy + iy is everywhere continuous but is not analytical.

 

4. Show that:

                

5. Prove that:

          ∫C     

where C is given by the equation | Z – a | = R.

 

6. Show that, if a function f (z) is continuous on a contour C of length l, and if M be the upper bound of $(z) on C, then:

| òc f(z)dz | £ ML.

7. Expand in the series the function :

          in the region |z| < 1.

 

Section—B

Note : Attempt any three questions. Each question carries 15 marks.

8. (i). Show that  does not exist.

    (ii) Show that

9. If ò (z) = u + iv is an analytic function of

Z= x + if and u-v =

find f(z) subject to the condition

10. State and prove Cauchy's integral formula.

11. State and prove Liouville's theorem.

12. State and prove Taylor's theorem.

13. Find the Taylor's and Laurent's series which represent the function :

                    

(i)                When | z |<2

(ii)             When 2< | z | < 3

(iii)            When | z |> 3.

 


 

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