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# Special Functions

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## ahmed sharaf

on 30 November 2016

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#### Transcript of Special Functions

S
pe
c
ia
l

F
un
c
tion
s

Special
Functions
G
amma
B
eta
B
essel
L
egender
H
yper
G
eometric
E
lliptic
R
iemann

Z
eta
Sherif
Taman
Shimaa & Eman
Nagwa & Aya
Eissa
G
amma &
B
eta
D
efinition
G
amma
Gamma & Beta functions were discovered by
l.Euler
and are referred also as
Euler’s
integrals of the second and first kind respectively.
is defined by the improper integral.
B
eta
Relation
G
&
B
E
lliptic
D
efinition
Let ω1 and ω2 be two complex numbers whose ratio is not real. Then a function which satisfies

For all values of z for which f (z) is defined is known as a doubly periodic function of z with periods ω1 and ω2.

A doubly periodic function that is analytic except at its poles, and which has no singularities other than these poles in a finite part of the complex plane is called an
elliptic
function.

P
roperties
Theorem 1.1.
Theorem 1.2.
Theorem 1.3.
Theorem 1.4.
Theorem 1.5.
Theorem 1.6.
Theorem 1.8.
Liouville’s theorem

D
efinition
Jacobi Elliptic Functions
P
roperties
Theorem 2.1.
Theorem 2.3.
Theorem 2.2.
A
pplications
Greenhill's Pendulums
Halphen's Circles and
Poncelet's Polygons
Fagnano's Ellipses
Spherical Trigonometry
Surface Area of an Ellipsoid
Seiffert's Spherical Spiral
Weierstrass Elliptic Functions
D
efinition
The Weierstrass zeta and sigma functions
Theorem 3.9.
Theorem 3.10.
A
pplications
The Spherical Pendulum
The Nine Circles Theorem
R
iemann
Z
eta
D
efinition
Defined in terms of multiple integrals
Defined in the complex plane by the contour integral
H
urwitz
Z
eta
F
unction
P
olylogarithm

L
erch
T
ranscendent
C
lausen
A
pplications
The
zeta
function occurs in applied statistics (see
Zipf's
law and
Zipf–Mandelbrot
law).

In one notable example, the
Riemann zeta
-function shows up explicitly in the calculation of the
Casimir
effect.

The
zeta
function is also useful for the analysis of dynamical systems.

L
egender
E
quation
R
odrigue's
F
ormula
O
rthognality
F
ourier
L
egender
E
xpansion
E
xample

A
pplication
H
yper
G
eometric
D
efinition
Is a special function represented by the
hyper geometric
series, that includes many other special functions as special or limiting case.
It is a solution of a second-order linear ordinary differential equation (
ODE
). Every second-order linear
ODE
with regular singular points can be transformed into this equation.
F
robenius
M
ethod
Solve
hyper

geometric
differential equation using
Frobenius
method.
it uses the series solution for a differential equation.
-Determine singular point.

-Determine p(z),q(z)-

-Using indicial equation

-Determine y(z),y'(z),y''(Z)

- Substitution into the differential equation

-This leads to the recurrence relation

-Substitution for the coefficients and write a solution

S
pecial
C
ases

The confluent
hyper

geometric
function
Legender
functions are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the
hyper

geometric
function." The solution of the
hyper

geometric
differential equation is very important.
For instance,
Legender
's differential equation can be shown to be a special case of the
hyper

geometric
differential equation"
L
egender
F
unction
Bessel
functions, can be expressed as limits of
hyper

geometric
functions.
These include most of the commonly used functions of mathematical physics.
B
essel
F
unction
A "cell" of an elliptic function.

Suppose we wish to count the number of poles or zeroes in a given period parallelogram.

We can simplify this calculation by translating the period parallelogram without rotation until no pole or zero lies on its boundary.

This parallelogram is known as a cell, and the set of poles or zeroes within that cell is called an irreducible set.

The number of poles of an
elliptic
function f(z) in any cell is finite.
The number of zeroes of an
elliptic
function f(z) in any cell is finite
The sum of the residues of an
elliptic
function f(z) at its poles in any cell is zero

An
elliptic
function f(z) of order m has m zeroes in each cell
If f(z) and g(z) are
elliptic
functions with poles at the same points and with the same principal parts at these points, then f(z) = g(z) + a, for some constant a.
If f (z) and g(z) are
elliptic
functions with zeroes and poles of the same order at the same points, then f (z) = ag(z), for some constant a.
The Jacobi elliptic function sn u is defined by means of the integral

For some constant k.

Therefore, by inversion of the integral, we have x = sn u.

It is clear that sn0 = 0.

The functions cn u and dn u are defined by the identities

It follows that cn0 = 1 = dn 0.

Each of the Jacobi elliptic functions depend on a parameter k, called the modulus.

We also have the complementary modulus k0 defined by

The function sn u is an odd function of u, while cn u and dn u are even functions of u.
The derivatives of the Jacobi elliptic functions are
The functions sn u and cn u each have a period 4K, while dn u has a smaller period 2K.
B
essel
Second order differential equation
considered one of the special functions

heat equation on a circle : cauchy-Euler

heat equation on a cylinder : Bessel function

B
ackground
essel

B
essel's
D
ifferential
E
quation
Recognize this as :

second-order differential equation
Variable coefficient

standard form
Devide entire equation by x to get

2
Any values of X will cause difficulty ?
X = 0

Singular point because 1/0 is not finite

the rest are ordinary points

F(x) and its derivatives exist at these ordinary points
what we know so far ?

there are solutions well behaved near X=0

AND Solutions that are singular at X=0
J (x) = Bessel function of the first kind of order n

well behaved solution near z=0

y (x) = Bessel function of the second kind of order n

singular solution at z=0

T
wo
T
ypes
O
f
S
olution

n
n
G
eneral
ٍٍS
olution
O
f
B
essel
D
ifferential
E
quation

P
roperties
O
f
B
essel

M
odified
B
essel
f
unctions
o
f
f
irst
a
nd
s
econd
k
ind

The modified Bessel's differential equation is defined in a similar manner by changing the variable X To ix (purely imaginary) in Bessel's differential equation
Its general solution is

where

G
raphs
o
f
B
essel
f
unctions
The Weierstrass zeta function ζ (z) is an odd function of z.
The Weierstrass sigma function σ (z) is an odd function of z.
A
pplications
I
ntroduction
Two types

1^(st )kind and 2^ndkind.

Especially important for problems related to

1-Wave propagation
2-Static potentials

Three important categories

1-Em waves in a cylindrical waveguide , heat conduction
2-Electronics and signal processing
3-Modes of vibration of an artificial membrane , acoustics

EM w
ave
i
n
c
ylindrical
w
aveguide
A
pplications
i
n
c
alculating
l
A
pplications
i
n
E
lectronics
a
nd
s
ignal
p
rocessing
A
pplications
i
n
a
coustics
(such as a drum or other membranophone)

A
pplications
E
lementary
F
unctions
In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots).
E
rror
F
unction
The error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion.

L
aguerre
P
olynomials
Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation:

Which is a second-order linear differential equation.
This equation has nonsingular solutions only if n is a non-negative integer.
More generally, the name Laguerre polynomials is used for solutions of

The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form
Hermite polynomials are a classical orthogonal polynomial sequence.

• probability, such as the Edge worth series;

• in combinatory, as an example of an Appell sequence, obeying the umbral calculus;

• in numerical analysis as Gaussian quadrature;

• in finite element methods as shape functions for beams;

• in physics, where they give rise to the eigenstates of the quantum harmonic oscillator;

• In systems theory in connection with nonlinear operations on Gaussian noise.

H
ermite
P
olynomials
Q
uantum
P
hysics
P
otenttial
E
nergy
M
ultipole
T
ringometry
A GREAT EXAMPLE THE ONE ELECTRON ATOM
The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge
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