MTH167: Complex Analysis and Transforms - Mathematics Assignment Help

Assignment Task:

Task:

Instructions:

1. Write all the answers in neat and clean handwriting. Question numbers should be correct.

2. All the answers should be well explained. Incomplete or logics used without reasons will get less credit. In other words, if you give correct answer but not proper logic, you will get only partial credits.

3. On each sheet there should be sheet number and Registration number of the student. On the first page, write number of sheets used.

4. Scan the sheets as one PDF. Upload on UMS.

5. The assignments received after the last date will not be considered.

6. Questions 1 to 4 are of 5 marks each and Questions 5 to 7 are of 10 marks each.

7. All the questions are compulsory.

Set -1

1) Prove or disprove: The function ????(????) = |2????| is analytic at origin. Give detailed reason in support of your answer.

2) Let real part of an analytic function ????(????) is equal to c (a constant). Find ????(????).

3) Evaluate ∫ ???? 2 ???????? from 1 + ???? to 2(1 + 2????) along the curve ???? = ???? 2 .

4) Expand 1 ????−5 for |????| < 5.

5) Prove that ???? = ???? 3 − 3???????? 2 . Prove that ????(????, ????) is harmonic and find v such that ????(????) = ???? + ???????? is analytic function.

6) Evaluate ∫(???? 2 + 2????) ????????, over the closed curve C, where C: Upper half of the circle |???? − 1| = 1.

7) Expand ????(????) = ???? (????+1)(????+2) about ???? = 2.

Set-2

1) Prove or disprove: The function ????(????) = |????| 2 is analytic at origin. Give detailed reason in support of your answer.

2) Let the imaginary part of an analytic function ????(????) is equal to c (a constant). Find ????(????).

3) Evaluate ∫(???? 2 + ????????????) ???????? from ????(1, 1) to ????(2, 4) along the curve ???? = ???? 2 .

4) Expand cos ???? about ???? = ????/4 .

5) Prove that ???? = 2????(???? + 1) − 4. Prove that ????(????, ????) is harmonic and find ???? such that ????(????) = ???? + ???????? is analytic function.

6) Evaluate ∫(???? 3 + 2) ????????, over the closed curve C, where C: Lower half of the circle |????| = 2.

7) Expand ????(????) = ???? ???? 2−5????+6 in the region 2< |????| < 3.

Set-3

1) Prove or disprove: A function which satisfies Cauchy Riemann equations is always analytic.

2) Let ????(????) = ???? + ???????? be an analytic function such that ???? + ???? = 5, find ????(????).

3) Evaluate ∫ ????? 2 ???????? from ????(0, 0) to ????(2, 1) along the curve ???? = 2????.

4) Expand log(2 + ????) about ???? = 0.

5) Prove that ???? = − sin ???? ?????????????????. Prove that ????(????, ????) is harmonic and find u such that ????(????) = ???? + ???????? is analytic function.

6) Evaluate ∫ 1 ???? ????????, over the closed curve C, where C: Upper half of the circle |????| = 1.

7) Expand ????(????) = 1 ???? 2+3????+2 about ???? = −2.

Set-4

1) Prove or disprove: Let ????(????) = ???? + ???????? be a function such that u and v are harmonic conjugates. Then ????(????) is analytic.

2) Let ????(????) = ???? + ???????? be an analytic function such that ???? − ???? = 3, find ????(????).

3) Evaluate ∫(???? + 2) ???????? from ????(0, 0) to ????(1, 1) along the curve ???? = ????.

4) Expand ???? ???? about ???? = 0.

5) Prove that ???? = ???? ???? cos ????. Prove that ????(????, ????) is harmonic and find v such that ????(????) = ???? + ???????? is analytic function.

6) Evaluate ∫ 1 ???? ????????, over the closed curve C, where C: Lower half of the circle |????| = 1.

7) Expand ????(????) = 1 ???? 2−3???? in the region 0 < |????| < 3

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