# B.Sc Mathematics Sample Paper for 3rd year Complex Analysis

**BACHELOR OF SCIENCE**** (MATHEMATICS), YEAR-3**

**COMPLEX ANALYSIS**

**Time: 3 hours M.Marks:60**

Note: – Attempt any 5 questions.

All questions carry equal marks.

**1.** **(a) **Write a short note on stereo-graphic projection.

** (b)** Find the complex number represented by the given point where *S* is the unit sphere with the centre origin.

**(c) **Derive the C-R equations in polar coordinates.

**2. (a) **Prove that every power series represents an analytic function at all points within the circle of convergence and its derivatives can be obtained by term wise differentiation of the given power series.

** ****(b) **Discuss the mapping.

**3.** **(a) **If is a function, which is analytic inside and on a simple closed curve C and if is any point in the interior of C, prove that .

**(b)** Evaluate *dz* where *C* is positively oriented circle.

**4. (a) **State and prove Liouville’s theorem.

**(b) **State and prove Morera’s theorem.

**(c) **Expand in aTaylor’s series about and determine the region of convergence.

**5. (a) **State and prove Laurent’s theorem.

**(b) ** Determine and classify the singular points of .

**6.** **(a) **State and prove Mittag-Leffler theorem.

**(b)** If *f* is an entire function and where *A* and *B* are constants, and if *f* have zeros with then prove that

**7. (a)** State and prove Rouche’s theorem.

**(b) **State and prove Legendre’s duplication formula.

**8.** **(a)** Show that any elliptic function with period can be written as where *C* is a constant.

**(b)** State and prove Cauchy’s Residue theorem.

Note: Above B.Sc Mathematics Sample Paper for 3rd year Complex Analysis has been prepared after reviewing few universities sample papers. We are not sure whether same or similar question papers pattern does your university has or not. We are just trying to help you by providing the possibilities. It is advisable to go thoroughly textbooks, teacher’s notes according to your syllabus for better preparation for your coming examination.

All the Best for your examination!!!

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