COMPLEX ANALYSIS

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.

The theory of functions of a complex variable, is the branch of mathematical analysis that investigates function of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering and electrical engineering. Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace’s equation, complex analysis is widely applicable to two-dimensional problems in physics.

L1: De Moivre’s Theorem

L2: Roots of a Complex Number

L3: Expansions of Sinnq, Cosnq and Tannq in Powers of Sinq, Cosq and Tanq, Respectively

L4: Expansion of Sinmq, Cosnq or SinmqCosnq in a Series of Sines or Cosines of Multiples of q

L5: Exponential and Circular Functions of a Complex Variable

L6: Hyperbolic Functions and Inverse Hyperbolic Functions – Relation Between Hyperbolic and Circular Functions – Formulae of Hyperbolic Functions – Inverse Hyperbolic Functions

L7: Real and Imaginary Parts of Circular and Hyperbolic Functions

L8: Logarithmic Function of a Complex Variable

L9: Summation of the Series ‘C + iS’ Method

L1: Limit, Continuity, Differentiability, Cauchy Riemann Equations – Analytic Functions – Applications of Flow Problems

L2: Theory: Geometrical Representations of w =f(z) – Conformal Mapping – Translation – Magnification and Rotation – Inversion and Reflection – Bilinear Transformation

L3: Problems: Geometrical Representations of w= f(z) – Conformal Mapping – Translation – Magnification and Rotation – Inversion and Reflection – Bilinear Transformation

L4: Complex Integration [Line integral of f(z)]

L5: Cauchy’s Theorem – Cauchy’s Integral Formula – Morera’s Theorem

L6: Taylor’s and Laurent’s Series

L7: Zeros and Singularities of Analytic Functions

L8: Residue Theorem and Calculation of Residues

L9: Evaluation of Real Definite Integrals