# B.Sc Statics 2nd year Sample Paper of Analysis Maths

** ****B.SC. STATISTICS, YEAR-2**** **

**ANALYSIS MATHS**** **

### Duration -3 Hours Max Marks: 60

**Note: 1. Attempt any FIVE questions. **

** 2. All questions carry equal marks.**

1. (a) Show that function : f(z) = (z ¹ 0), f(0) = 0 is not analytic at z = 0, although C.R. equations satisfied at that point.

(b) Derive the formula for the radius of convergence of the power series a_{n }z^{n} f find the same for the series.

2. (a Using Cauchy’s integral formula, prove that where C is the circle |z| = 3

(b) State & prove weiestrass theorem concerning the behaviour of an analytic function near an isolatial essential singularity.

3. (a) State & prove Argument principle

(b) Prove that every polynomial of degree n has no zeros & determine the number of roots of the equation

z^{8} – 4z^{5} + z^{2} – 1 = 0 that lies inside the circle |z| = 1

4. (a) State & prove minimum modulus principle.

(b) Show that cot z =

5. (a) State & proveTaylor’s Theorem.

(b) Expand f(z) = in Lament’s series valid for the regions

| < |z+1| < 2

6. (a) State & prove Fundamental theorem of Algebra with the help of Morera theorem.

(b) Expand log |1+z| isTaylor’s series about z = 0 & determine the region of convergence for the resulting series

7. (a) There cannot be two different direct analytic continuations of a function

(b) If f(z) is memorphic inside a closed contour c & has no zero on c then

= N-P

N is the number of zeros & P the number of poles inside C.

8. (a) State & prove Rouche’s theorem,

(b) State & prove Mittag-Leffler Theorem.

NOTE: Above sample paper** **for B.Sc Economics 2^{nd} year of Analysis Maths has** **been created after reviewing various universities question papers. We are not sure whether your university has same or similar question pattern. We always suggest you to go thoroughly with text books, teachers’ notes etc. for better results.

We wish you best of luck for upcoming examinations.

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