A complex number z is an ordered pair (x, y) of real numbers

The theory of integral transforms affords mathematical devices through which solutions of
numerous boundary value problems of engineering can be obtained e.g. conduction of heat,
transverse vibrations of a string, transverse oscillations of an elastic beam, transmission lines
etc.
The choice of a particular transform to be employed for the solution of an equation
depends on the boundary conditions of the problem and the ease with which the transform can
be inverted.
An integral transform when applied to a partial differential equation reduces the number
of its independent variables by one.

Consider the equation of the form f(x) = 0
If f(x) is a quadratic, cubic or biquadratic expression then algebraic formulae are available
for expressing the roots. But when f(x) is a polynomial of higher degree or an expression involving
transcendental functions e.g., 1 + cos x – 5x, x tan x – cosh x, e–x – sin x etc., algebraic methods
are not available.
Here, we shall describe some numerical methods for the solution of f(x) = 0, where f(x) is
algebraic or transcendental.

Consider a function of a single variable y = f(x). If f(x) is defined as an expression, its derivative
or integral may often be determined using the techniques of calculus.
However, when f(x) is a complicated function or when it is given in a tabular form, we
use numerical methods.

An operation which combines two elements of a set to give another element of the same set is
called a binary operation or a binary composition.

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B. Tech. [SEMESTER-IV]
THEORY EXAMINATION, 2012–2013
MATHEMATICS—III