**Complex Variables: **Functions of complex
variables, continuity( only statement ), derivability of a function, analytical
regular function, necessary condition for a function to be analytic, statement
of sufficient conditions, Cauchy Riemann equations in polar co-ordinates.
Harmonic functions, orthogonal trajectories, Analytical and Milne Thomson method
to find f(z) from its real or imaginary part.

**Mapping :** conformal mapping, linear and
bilinear mapping with geometrical interpretations.

**Fourier Series and Integrals : **Orthogonal
and orthonormal functions, expression of a function in a series of orthogonal
functions, sine and cosine functions and their orthogonality properties. Fourier
series, Drichlet conditions ( only statement ), periodic functions, even and odd
functions, half range sine and cosine series,Parsevals relation, Complex form of
Fourier series, introduction to Fourier integral, relation with Laplace
transform.

**Laplace Transforms: **Function of bounded
variable ( statement only ), Laplace transforms of 1, at, exp( at ), sin( at ),
cos( at ), sinh( at ), cosh( at ), erf( t ), shifting properties, expressions
with proofs for L { t f(t) }, L {f(t)/t }, Laplace of an integral and
derivative. Unit step functions, Heavyside, Dirac Delta functions and their
Laplace transform, Laplace transform of periodic functions.Evaluation of inverse
Laplace transforms, partial fraction method, Heavyside development, Convolution
theorem. Application to solve initial and boundary value problems involving
ordinary differential equations with one variable.

**Matrices : **Types of matrices, adjoint of a
matrix, inverse of a matrix, elementary transformations, rank of a matrix,
linear dependent and independent rows and columns of a matrix over a real field,
reduction to a normal form, partitioning of matrices.System of homogenous and
non homogenous equations, their consistency and their solutions.