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Classical Quantum & Statistical Physics
1.
Evaluation of integrals using gamma functions?
The gamma function in the complex plane
The behavior of Γ(z) for an increasing positive variable is simple: it grows quickly — faster than
an exponential function. Asymptotically as
, the magnitude of the gamma function is
given byStirling's formula
where the symbol ~ means that the quotient of both sides converges to 1.
The behavior for nonpositive z is more intricate. Euler's integral does not converge for z ≤ 0,
but the function it defines in the positive complex half-plane has a unique analytic
continuation to the negative half-plane. One way to find that analytic continuation is to use
Euler's integral for positive arguments and extend the domain to negative numbers by
repeated application of the recurrence formula,
choosing n such that z + n is positive. The product in the denominator is zero
when z equals any of the integers 0, −1, −2,... . Thus, the gamma function must be
undefined at those points due to division by zero; it is a meromorphic
function with poles at the nonpositive integers. The following image shows the graph of
the gamma function along the real line:
The gamma function is nonzero everywhere along the real line, although it comes
arbitrarily close as
. There is in fact no complex number z for which Γ(z) = 0,
and hence the reciprocal gamma function 1 / Γ is an entire function, with zeros
at z = 0, −1, −2,.... We see that the gamma function has a local minimum
at
where it attains the value
. The
gamma function must alternate sign between the poles because the product in the
forward recurrence contains an odd number of negative factors if the number of poles
between z and z + n is odd, and an even number if the number of poles is even.
2.
Discuss the Properties and related theorem for Inverse Laplace Transform?
In mathematics and with many applications in physics and engineering and throughout the
sciences, the Laplace transform is a widely used integral transform. Denoted
, it is
a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms it to a function F(s)
with a complex argument s. This transformation is essentially bijective for the majority of practical
uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the
useful property that many relationships and operations over the originals f(t) correspond to
simpler relationships and operations over the images F(s).[1] It is named for Pierre-Simon
Laplace, who introduced the transform in his work on probability theory.
The Laplace transform is related to the Fourier transform, but whereas the Fourier transform
expresses a function or signal as a series of modes of vibration (frequencies), the Laplace
transform resolves a function into its moments. Like the Fourier transform, the Laplace transform
is used for solving differential and integral equations. In physics and engineering it is used for
analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical
devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a
transformation from thetime-domain, in which inputs and outputs are functions of time, to
the frequency-domain, where the same inputs and outputs are functions of complex angular
frequency, in radians per unit time. Given a simple mathematical or functional description of an
input or output to a system, the Laplace transform provides an alternative functional description
that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new
system based on a set of specifications.
3.
Explain Convolution theorem and deduce the Orthogonally of Pn(x)?
In mathematics, the convolution theorem states that under suitable conditions the Fourier
transform of a convolution is the pointwise product of Fourier transforms. In other words,
convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain
(e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related
transforms. Let and be two functions with convolution
. (Note that the asterisk denotes
convolution in this context, and not multiplication. The tensor product symbol
is sometimes
used instead.) Let
Fourier transforms of
denote the Fourier transform operator, so
and
and
are the
, respectively. Then
where denotes point-wise multiplication. It also works the other way around:
By applying the inverse Fourier transform
, we can write:
Note that the relationships above are only valid for the form of the Fourier transform
shown in the Proof section below. The transform may be normalised in other ways,
in which case constant scaling factors (typically
relationships above.
or
) will appear in the
This theorem also holds for the Laplace transform, the two-sided Laplace
transform and, when suitably modified, for the Mellin transform and Hartley
transform (see Mellin inversion theorem). It can be extended to the Fourier transform
of abstract harmonic analysis defined over locally compact abelian groups.
This formulation is especially useful for implementing a numerical convolution on
a computer: The standard convolution algorithm has quadratic computational
complexity. With the help of the convolution theorem and the fast Fourier transform,
the complexity of the convolution can be reduced to O(n log n). This can be
exploited to construct fast multiplication algorithms.
4.
Discuss the Expansion of periodic functions in Fourier series
Definition
A function f is said to be periodic with period P if (for some nonzero constant P) we have
for all values of x. If there exists a least positive[1] constant P with this property, it is called
the prime period. A function with period P will repeat on intervals of length P, and these
intervals are sometimes also referred to as periods.
Geometrically, a periodic function can be defined as a function whose graph
exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph
of f is invariant undertranslation in the x-direction by a distance of P. This definition of periodic
can be extended to other geometric shapes and patterns, such as periodic tessellations of
the plane.
A function that is not periodic is called aperiodic.
[edit]Examples
For example, the sine function is periodic with period 2π, since
for all values of x. This function repeats on intervals of length 2π (see the graph to the
right).
Everyday examples are seen when the variable is time; for instance the hands of
a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in
which the position(s) of the system are expressible as periodic functions, all with
the sameperiod.
For a function on the real numbers or on the integers, that means that the
entire graph can be formed from copies of one particular portion, repeated at regular
intervals.
A simple example of a periodic function is the function f that gives the "fractional part" of
its argument. Its period is 1. In particular,
f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5.
The graph of the function f is the sawtooth wave.
A plot of f(x) = sin(x) and g(x) = cos(x); both functions are periodic with period 2π.
The trigonometric functions sine and cosine are common periodic functions, with
period 2π (see the figure on the right). The subject of Fourier series investigates the
idea that an 'arbitrary' periodic function is a sum of trigonometric functions with
matching periods.
According to the definition above, some exotic functions, for example the Dirichlet
function, are also periodic; in the case of Dirichlet function, any nonzero rational
number is a period.
[edit]Properties
If a function f is periodic with period P, then for all x in the domain of f and all
integers n,
f(x + nP) = f(x).
If f(x) is a function with period P, then f(ax+b), where a is a positive constant, is
periodic with period P/a. For example, f(x)=sinx has period 2π, therefore sin(5x)
will have period 2π/5.
[edit]Double-periodic
functions
A function whose domain is the complex numbers can have two
incommensurate periods without being constant. The elliptic functions are such
functions. ("Incommensurate" in this context means not real multiples of each
other.)
5.
Explain the Full wave rectifier?
A rectifier is an electrical device that converts alternating current (AC), which periodically
reverses direction, to direct current (DC), which flows in only one direction. The process is known
as rectification. Physically, rectifiers take a number of forms, including vacuum
tube diodes, mercury-arc valves,solid-state diodes, silicon-controlled rectifiers and other siliconbased semiconductor switches. Historically, even synchronous electromechanical switches and
motors have been used. Early radio receivers, called crystal radios, used a "cat's whisker" of fine
wire pressing on a crystal of galena (lead sulfide) to serve as a point-contact rectifier or "crystal
detector".
Rectifiers have many uses, but are often found serving as components of DC power
supplies and high-voltage direct current power transmission systems. Rectification may serve in
roles other than to generate direct current for use as a source of power. As
noted, detectors of radio signals serve as rectifiers. In gas heating systems flame rectification is
used to detect presence of flame.
The simple process of rectification produces a type of DC characterized by pulsating voltages and
currents (although still unidirectional). Depending upon the type of end-use, this type of DC
current may then be further modified into the type of relatively constant voltage DC
characteristically produced by such sources as batteries and solar cells.
Rectifier devices
Before the development of silicon semiconductor rectifiers, vacuum tube diodes and copper(I)
oxide or selenium rectifier stacks were used. High power rectifiers, such as are used in highvoltage direct current power transmission, now uniformly employ silicon semiconductor devices of
various types. These are thyristors or other controlled switching solid-state switches which
effectively function as diodes to pass current in only one direction.
6.
What are Recurrence relations? Explain Expansion of Jn(x) when n is half an
odd integer?
difference equation" redirects here. It is not to be confused with differential equation.
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once
one or more initial terms are given: each further term of the sequence is defined as a function of
the preceding terms.
The term difference equation sometimes (and for the purposes of this article) refers to a specific
type of recurrence relation. However, "difference equation" is frequently used to refer
to any recurrence relation.
An example of a recurrence relation is the logistic map:
with a given constant r; given the initial term x0 each subsequent term is determined by this
relation.
Some simply defined recurrence relations can have very complex (chaotic) behaviours, and
they are a part of the field of mathematics known as nonlinear analysis.
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive
function of n.
The Fibonacci numbers are the archetype of a linear, homogeneous recurrence relation with
constant coefficients (see below). They are defined using the linear recurrence relation
with seed values:
Explicitly, recurrence yields the equations:
etc.
We obtain the sequence of Fibonacci numbers which begins:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
It can be solved by methods described below yielding the closed
form expression which involve powers of the two roots of the
characteristic polynomial t2 = t + 1; the generating function of the
sequence is the rational function
7.
Explain the Basic theorem of Laplace transforms and its properties?