Complex Analysis
Paper Code:
MAT601
Credits:
3
Contact Hours:
45.00
Objective:
o To improve analytical skill.
o To improve basics in mathematics.
Unit I:
I
9.00
Complex Plane, connected and compact sets, statement of Jordan curve theorem. Extended complex plane: Stereographic projection. Complex valued functions, Analytic functions, Cauchy-Riemann equations (Cartesian & Polar forms), Harmonic functions. Construction of an analytic function.
Unit II:
II
9.00
Complex Integration, Complex line Integrals, Cauchy’s Integral theorem, Cauchy’s Fundamental theorem, Indefinite integrals, Fundamental theorem of Integral Calculus for complex functions, Cauchy’s Integral Formula, Analyticity of the derivative of an analytic function.
Unit III:
III
9.00
Liouville’s theorem, Poisson’s Integral formula, Morera’s theorem . Maximum modulus principle. Taylor’s and Laurent’s Series.
Unit IV:
IV
9.00
Singularities. Meromorphic functions and Entire functions. Riemann’s theorem, Casorati- Weirstrass theorem. Rouchy’s theorem. Fundamental theorem of algebra. Cauchy’s Residue theorem.
Unit V:
V
9.00
Contour integration. Conformal mapping (Introduction). Bilinear transformation and their simple properties. Power series.
Essential Readings:
- Complex Analysis, Purohit and Goyal, Jaipur Publishing House
- Complex Variables: Theory and Applications, H.S. Kasana, Prentice Hall, Delhi
- Foundation of Complex Analysis, S. Ponnuswamy, Narosa Publishers.
- Complex Variables and Application, Brown and Churchill, McGraw Hills Book Co.
References:
- Theory and Problems of Complex Variables, R. Murray Spiegel, Schaum Outline Series
Academic Year:
