Complex number system, Complex number – Field of Complex numbers –
Conjugation – Absolute value – Arguments Simple Mappings.
i) w= z+a ii) w= az iii) w=1/z
invariance of cross-ratio under bilincar transformation-Definition of extended complex planc –
Complex functions : Limit of a function – continuity – differentiability- Analytical
function defined in a region – necessary conditions for differentiability – sufficient conditions for
differentiability – Cauchy- Riemann equation in polar coordinates- Definition of entire function.
Power Series : Absolute convergence – circle of convergence – Analyticity of the sum
of power series in the Circle of convergence ( term by term differentiation of a series)
Elementary functions : Exponential, Logarithmic, Trigonometric and Hyperbolic functions.
Anx.18 D - B.Sc Maths (SDE) 2007-08 Page 10 of 26
Conjugate horonic functions : Definition and determination, Conformal Mappings :
Isogonal mapping – Conformal mapping- Mapping z --- f (z),where f is analystic, particularly the
w=ez ; w =z ½ ; w=sin z ; w=1/2 (z+1/z)
Complex Integration : Simply and multiply connected regions in the complex plane.
Integration of f(z) from definition along a curve joining Z1 and Z2 . Proff of Cauchy’s Theorem
( using Goursat’s lemma for a simply connected region). Cauchy’s integral formula for higher
derivatives ( statement only) – Morera’s theorem.
Results based on Cauchy’s theorem (I) : Zero-Cauchy’s Inequality – Lioville’s theorem –
Fundamental theorem of algebra Maximum modulus theorem Gauss mean value theorem Gauss
mean value theorem for a harmonic function on a circle.
Results based on Cauchy’s theorem (II) – Taylor’s series-Laurent’s series.
Singularities and Residues: Isolated singularities ( Removable Singularity, pole and
essential singularity ) Residues Residue theorem.
Real definite integrals : Evaluation using the calculus of residues- Integration on the unit circle-
Integral with - and as lower and upper ,limits with the following integrals:
I. P(x) / Q(x) where the degree of Q(x) exceeds that of P(x) at least.
II. (sin ax ).f(x),(cos ax).f(x), where a>0 and f(z) ---- 0 as z ----- and f(z) does not have a
pole on the real axis.
III. f(x) where f(z) has a finite number of poles on the real axis.
Integral of the type
Meromorphic functions : Theorem on number of zeros minus number of poles –
Principle of argument : Rouche’s throrem – theorem that a function which is mcromorphic in the
extended plane is a rational function.