Venue: |
BIM, Pune |

Dates: |
14 - 26 May 2007 |

Convener(s) |
Speakers, Syllabus and Time table |
Applicants/Participants |

Name |
A. R. Shastri (Academic) |
S. A. Katre (Local) |

Mailing Address |
Dept of Mathematics, |
Dept. of Mathematics, |

Advanced Training School in Mathematics for Lecturers (ATML) in Complex Analysis is being organised in Pune at Bhaskaracharya Pratishthana in May 2007 on behalf of NBHM.

** National Coordinating Committee**Director: Ravi S. Kulkarni

Secretary: J. K. Verma

Members : S. D. Adhikari, Satya Deo, S. A. Katre, Shobha Madan, I. B. S. Passi, Ravi A. Rao

#### Members of the Local Organising Committee

A. R. Shastri, IIT Bombay | (Academic Convener) |

S. A. Katre, Univ. Pune | (Local Coordinator) |

C. S. Inamdar, Pune | (Custodian, BP) |

R. V. Gurjar, TIFR | Hon. Res. Director, BP |

Manjusha Joshi, BP |

**The topics to be covered** (numbers in the bracket indicate the the number of lectures allotted):

**I- week**

1.1 Basic properties of complex numbers:historical remarks, why the name complex etc.(1)

2.1 Review of Diff. Calculus of 2-variables.(1)

1.2 Geometry of complex numbers: rigid motions of the plane.(1)

2.2 Complex differentiability; CR equations.(1)

1.3 Sequences and series:uniform convergence, Weierstrass' M-test.(1)

3.1 Line integrals, M-L inequality; diff. under integral sign(1)

1.4 Formal power series, convergent power series, analytic functions; Exponential, trigonometric, and logarithmic functions; complex exponents.(1)

3.2 Path-independence; exactness, existence of primitives.(1)

1.5 Extended complex plane: FLTs.(1)

3.3 Cauchy-Goursat for triangles; Cauchy's theorem, Integral formula, estimates, Liouville's theorem. (1)

1.6 Conformality: applications.(1)

3.4 FTA, Taylor's theorem and analyticity, Zeros of holomorphic functions; Identity theorem.(1)

**II-week**

3.5 Isolated singularities.(1)

4.1 Winding number.(1)

3.6 Laurent series and residues.(1)

4.2 Argument principle, local solutions, open mapping theorem.(1)

2.3+2.4 Simple connectivity, homology and homotopy form of Cauchy's theorem.(2) 4.3 Inverse function theorem, branch theorem, Maximum Modulus Principle, Schwartz lemma.(1)

4.4 Residues, GAP, Rouche's theorem.(1)

4.5+4.6 Applications to improper integrals.(2)

2.5+2.6 Convergence in function theory, Runge's theorem with some applications. (2)

**Special Lectures:**

1. Application of Complex Analysis in Fluid Dynamics. (VDS)

2. Gamma function, Riemann zeta function. (GKS)

3. Gaussian Sums. (MN)

** **

I-week:14th May to 19th May |
||||||

KDJ: 1.1-1.6; ARS: 2.1,2.2; SAK: 3.1-3.4 | ||||||

Day | Mon | Tue | Wed | Thu | Fri | Sat |

9.30-11.00 | KDJ | KDJ | KDJ | KDJ | KDJ | KDJ |

11.30-1.00 | ARS | ARS | SAK | SAK | SAK | SAK |

1.00-2.30 | Lunch Break |
|||||

2.30-4-30 | Tutorial | Tutorial | Tutorial | Tutorial | Tutorial | Tutorial |

II-week:21th May to 26th May |
||||||

SAK: 3.3-3.6; HB: 4.1-4.6; ARS: 2.3-2.6 | ||||||

Day | Mon | Tue | Wed | Thu | Fri | Sat |

9.30-11.00 | HB | HB | HB | HB | HB | HB |

11.30-1.00 | SAK | SAK | ARS | ARS | ARS | ARS |

1.00-2.30 | Lunch Break |
|||||

2.30-4-30 | Tutorial | Tutorial | Tutorial | Tutorial | Tutorial | Tutorial |

5.00-6.00 | VDS | GKS | GKS | MN |

**Resource Persons:**

**Speakers**

Prof. K. D. Joshi kdj *at*math.iitb.ac.inProf. S. A. Katre sakatre *at*math.unipune.ernet.inDr. H. Bhate hbhate *at*math.unipune.ernet.inProf. A. R. Shastri ars *at*math.iitb.ac.in**Invited Speakers**

V. D. Sharma, IIT Bombay: Application of Complex Analysis in Fluid Dynamics

Gopal Srinivasan, IIT Bombay: Gamma function, Riemann zeta function

Mangala Narlikar, BIM: Gaussian Sums**Course Associates**

1. Sameer Chavan, IISER, Pune. sameer_chavan95 *at*yahoo.co.in2. Pratul Gadagkar, Univ. of Pune, Pune pratul1980 *at*math.unipune.ernet.in3. Vinay Wagh, IIT, Bombay, Mumbai vinay *at*math.iitb.ac.in

**Syllabus**

- Basic algebraic and geometric properties of complex numbers, rigid motions. Open sets, Connectivity and path connectivity.
- Complex differentiability, CR-equations, holomorphic anti-holomorphic functions, operators harmonic functions.
- Review of sequences and series, Uniform convergence, Weierstrass' M-test, power series, Analytic functions, exponential and trigonometric functions logarithmic function, complex exponents of complex numbers.
- Contour integration, existence of primitives, Cauchy-Goursat theorem, winding number, Cauchy's theorem, Cauchy's integral formulae, Morera's theorem.
- Local behaviour of Analytic functions, Taylor series. zeros of analytic functons, identity theorem, Analysis of isolated singularities. Poles and residues. Laurent Expansion, Casorati Weierstrass' theorem. Open mapping theorem. Maximum modulus principle. Schwarz' lemma.
- Runge's Approximation Theorem, Generalization of Cauchy's theorem, Argument principle, simple connectivity, Residue theorem, Integral formula for Inverse function, Holomorphicity of Symmetric functions.
- Computation of real integrals.
- Convergene in function Theory, Hurwitz theorem, Mittag-Leffler theorem, partial fraction development, infinite products, Weierstrass's theorem, Gamma-function and Riemann zeta-function.
- Normal families, Equicontinuity, Arzela-Ascoli, Montels theorem. Riemann mapping theorem.
- Harmonic functions, Poisson Integral, Schwarz's reflection principle, Harnack' principle, Sub-harmonicity, Dirichlet's problem, Greens's function, another proof of Riemann mapping theorem.

**References:**

- Ahlfors, V.,
*Complex Analysis*MacGraw-Hill, - Remmert, R.,
*Theory of Complex Functions*, GTM-122 Springer. - Rudin, W.,
*Real And Complex Analysis*, MacGraw-Hill 2nd ed. - Shastri A. R.,
*An introduction to Complex Analysis*, MacMillan India, 1999.

Selected Applicants |

Sr. No. |
Name |
Place |
Accommodation: H-Rooms at Guest House of BP, R-Rooms at main building of BP, Flat-is flat near BP. |

1. | Dr. Arshad Khan | Jamia Millia Islamia, New Delhi | H-3 |

2. | Priyabrat Gochhayat | Berhampur University, Orissa. | R-5 |

3. | S. Balaji | The M.D.T. Hindu College, Tirunelveli | H-2 |

4. | Phatangare Nanasaheb M. | New Arts, Comm. Sci. College, Ahmednagar, Maha. | Flat |

5. | Aher Bhalchandra Yadav | Arts, Comm. Sci. College, Ozar Mig, Nashik, Maha. | Flat |

6. | Ratan Kumar Dutta | Dept. of Math. Visva-Bharati, Santiniketan | R-5 |

7. | Saiful Rahman Mondal | IIT Roorkee | H-3 |

8. | Ms. Hemlata Pawar | Dr. D. Y. Patil Institute of Engineering and Technology, Pune | Local |

9. | Ms. Vandna Agnihotri | C.S.J.M. University, Kanpur | R-1 |

10. | Dr. Uday Hanmant Naik | Willingdon College, Sangli, Maha. | H-4 |

11. | Shivaji Ashok Tarate | N.A.C. Sc. College, Ahmednagar, Maha. | Flat |

12. | Bibhas Chandra Mondal | Surendranath College, Kolkata | R-5 |

13. | Trilok Mathur | Banasthali Vidyapith, Jaipur | H-2 |

14. | Faizan Ahmad Khan | Women's College, A.M.U. Aligarh | H-3 |

15. | Vinod I. Sharma | Sardar Patel College, Engineering, Mumbai, Maha. | R-5 |

16. | Pravin Garg | Banasthali Vidyapith, Newai Rajasthan | H-2 |

17. | Umesh | Rajdhani College, University of Delhi, Delhi | R-5 |

18. | Sachin D. Patil | Jaysingpur College, Jaysingpur | Flat |

19. | Ms. Sandhya Singhal | Deen Dayal Upadhaya College, New Delhi | R-1 |

20. | Dr. M. Marudai | The M.D.T. Hindu College, Tirunelveli | H-4 |

21. | S. Mathugnaniah | St. Xavier's College, Palayamkottai | H-4 |

22. | Patil Dattatraya | Sangamner college, Sangamner, Maha. | Flat |

23. | Dr. P. Sam Johnson | SSN College of Engineering, Chennai | Flat |

24. | Ms. Shinde Akanksha | K. T. H. M. College, Nasik, Maha. | R-2 |

25. | J. Maria Joseph | Arun Anandar College, Tiruvanna Malai | Flat |

26. | Ms. Asha L. Mangalani | C.H.N. College, Ulhas Nagar, Dist. Thane, Mah. | R-1 |

27. | Sanjay Kumar | IIT Bombay, Mumbai | R-5 |

28. | Karamjit Singh | B.P.R. College, Kuruksketra, Hariyana | R-5 |

29. | Javid Ali | A.M.U. Aligarh | H-3 |

30. | Jayanta Borah | Itanagar, Arunachal Pradesh | H-2 |

31. | Vinay Arora | S.S.H. Panjab U.R.C., Hoshiyarpur, Panjab | R-5 |

32. | Kuldeep Sharma | S.S.H. Panjab U.R.C., Hoshiyarpur, Panjab | R-5 |

33. | Ms. Manju Prasad | AISS M S women college of Engineering, Pune, Maha. | Local |

34. | Ms. Hurratulmalika Juzer Siamwalla | Abeda Inamdar Sr. College, Pune, Maha. | Local |

35. | Dr. Krishna Kumar | JSPM's Imperial College of Engg. Research, Wagholi, Pune, Maha. | Local |

How to reach |