Unit 1 Polar Form of Complex Numbers, Powers and Roots,

12 lecture hours

• Complex Variable – Differentiation : Differentiation
• Cauchy-Riemann equations
• analytic functions
• harmonic functions
• finding harmonic conjugate
• elementary analytic functions (exponential
• logarithm) and their properties
• Conformal mappings
• Mobius transformations and their properties

Unit 2 Complex Variable - Integration

8 lecture hours

• Contour integrals
• Cauchy-Goursat theorem (without proof)
• Cauchy Integral formula (without proof)
• Liouville’s theorem and Maximum-Modulus theorem (without proof)
• Convergence Tests
• Power Series
• Functions Given by Power Series
• Taylor and Maclaurin Series
• Uniform Convergence

Unit 3 Laurent’s series; Zeros of analytic functions, singularities, Residues, Cauchy Residue theorem

• (without proof)
• Residue Integration Method
• Residue Integration of Real Integrals. 06 14%

Unit 4 First order partial differential equations, solutions of first order linear and nonlinear PDEs,

• Charpit’s Method 06 14%

Unit 5 Solution to homogeneous and nonhomogeneous linear partial differential equations second and

10 lecture hours

• higher order by complementary function and particular integral method
• Separation of variables method to simple problems in Cartesian coordinates
• second-order linear equations and their classification
• Initial and boundary conditions
• Modeling and solution of the Heat
• Wave and Laplace equations