Bessel
Functions
One of the varieties of special functions which are
encountered in the solution of physical problems is the class of functions
called Bessel functions. They are solutions to a very important differential
equation, the Bessel equation: 

The solutions to this equation are in the form of
infinite series which are called Bessel functions of the first kind. The
expression for the sum is: 

Values for the Bessel functions can be found in most
collections of mathematical tables. Bessel functions are encountered in
physical situations where there is cylindrical symmetry. This occurs in
problems involving electric fields, vibrations, heat conduction, optical
diffraction and others. 
A specific class of special
functions called spherical Bessel functions arises in problems of
spherical symmetry. The first three forms are 

From the first form, the higher order forms can be
generated from the relationship 

It is sometimes useful to have the limiting cases for
these functions for very large or very small distances: 

The derivative has certain special properties when applied to combinations of functions. 

If the function f(x) is a product of two functions m(x) and n(x), then the derivative of this product is 

If the function f(x) is a sum of two functions m(x) and n(x), then the derivative of this sum is simply the sum of the derivatives: 

If the function y = f(x) and x=g(z), then the derivative of y with respect to z can be written as a product of derivatives: 
Definite Integrals Associated with Gaussian Distributions
In physical systems which can be modeled by a Gaussian distribution, one sometimes needs to obtain the average or expectation value for physical quantities. If these properties depend on x, then they can be integrated to find the average value. For the first five powers of x, the integrals have the following forms: 
Definite Integrals of the Squares of Trig Functions
The
form of the integral of square of a trig function is 

Form if you can evaluate
the following definite integrals which are useful in averaging sinusoidal
functions. No mater how you slice it as long as you take any number of exact
quarter intervals, the average of the square of a sin or cos is 1/2 The average of a function
over an interval is the integral of the function over that interval divided
by the interval if you divide any of the integrals to the left by their
intervals you get ½ 
First Order Nonhomogeneous Differential Equation
An example of a first order linear nonhomogeneous differential equation is 

Having a nonzero value for the constant c is what makes this equation nonhomogeneous, and that adds a step to the process of solution. The path to a general solution involves finding a solution to the homogeneous equation (i.e., drop off the constant c), and then finding a particular solution to the nonhomogeneous equation (i.e., find any solution with the constant c left in the equation). The solution to the homogeneous equation is 

By substitution you can verify that setting the function equal to the constant value c/b will satisfy the nonhomogeneous equation. 

It is the nature of differential equations that the sum of solutions is also a solution, so that a general solution can be approached by taking the sum of the two solutions above. The final requirement for the application of the solution to a physical problem is that the solution fits the physical boundary conditions of the problem. The most common situation in physical problems is that the boundary conditions are the values of the function f(x) and its derivatives when x=0. Boundary conditions are often called "initial conditions". For the first order equation, we need to specify one boundary condition. For example: 

Substituting at x=0 gives: 

Derivatives of Common Functions In this table, a is a constant, while u, v, w are functions. The
derivatives are expressed as derivatives with respect to an arbitrary
variable x. 
Indefinite Integrals of Common Algebraic Forms To each of these indefinite integrals should be added an arbitrary
constant of integration, which cannot be determined by the integration
process. 
The basic probability of throwing a "3" in one throw is p=1/6 . Then for n=6 throws, the average number of 3's would be a=np= 6 x 1/6 = 1. So throwing two 3's is definitely not the expected or most probable result, but it remains to be seen how much the probability is diminished relative to the most probable result. Now the number of ways to pick r objects out of a total of n is: n(n1)(n2)...(nr+1) = n!/(nr)! which is called the permutation. But considering them to be distinguishable, this overcounts. If getting the same objects or outcomes but in different order is not going to be considered to be a different overall outcome, then we need to divide by the number of ways to arrange them, which is r! . The number of ways to pick distinguishable sets is then n!/[r!(nr)!] which is called the combination.
The probability of getting a certain outcome r times is p^{r}, but if you do that AND get some other outcome for the remainder of the tries (associated probability (1p)^{nr}) then the probability for this combined outcome is the product of these probabilities. This is a part of the nature of general probability: the probability for any two outcomes related by a logical "AND" will be the product of the probabilities of the single outcomes. Then if the combined probability is multiplied by the number of ways to get this outcome, the result is the binomial distribution function.
A form of integral which shows up regularly in quantum
statistics is: 

The evaluation of this class of integrals is in terms of two special
functions, the gamma function and the Riemann zeta
function. 

A function which shows up often in statistical
calculations is the gamma function: 


This can be related to another integral 


leading to the relationships 


Some other common relationships with gamma functions: 


One of the applications of the gamma
function is in the evaluation of statistical integrals. 

To each of these indefinite integrals should be added an arbitrary
constant of integration, which cannot be determined by the integration
process. 
To each of these indefinite integrals should be added an arbitrary
constant of integration, which cannot be determined by the integration
process. The definite integrals of the squares of trig functions are of
interest in many physical problems. 

Integration by parts is a
useful strategy for simplifying some integrals. It is based on the
combination rule for differentiation and the general approach can be
summarized by: 

This technique is particularly appropriate for removing
a linear term multiplying an exponential. For example, the integral 

can be simplified by the identification 

Then u can be differentiated and dv can be integrated. 

The value of the integral is then 

If a higher power term of x
multiplies the exponential, then the process of integration by parts can be
repeated to reduce the power term. 
One of the varieties of
special functions which are encountered in the solution of physical problems
is the class of functions called Legendre polynomials. They are solutions to
a very important differential equation, the Legendre equation: 

The polynomials may be denoted by P_{n}(x) ,
called the Legendre polynomial of order n. The polynomials are either even or
odd functions of x for even or odd orders n. The first few polynomials are
shown below. 

The general form of a Legendre polynomial of order n is
given by the sum: 

An important class of special
functions called the associated Legendre functions can be derived from the
Legendre polynomials. The defining relationship is 

where P_{n}(x) is the Legendre polynomial of
order n. These functions are of great importance in quantum physics because
they appear in the solutions of the Schrodinger
equation in spherical polar coordinates. In that context the
variable x is replaced by cosq,
where q is the colatitude angle.
Also in that context, the wavefunctions which are the solutions of the
Schrodinger equation need to be normalized, so the list of functions below
will include the normalization factor 

The normalized functions are of the form 

and appear in the wavefunctions for the hydrogen atom.
The first few normalized associated Legendre functions are listed below. 

The associated Legendre functions can
be used to construct another important set of functions, the spherical harmonics. 
The Mean of the Binomial Distribution
The mean value of the
binomial distribution is a = np where n is the number of events and p is the
probability for each event. Binomial distribution 
Mean: np 
This seems a very simple expression for the mean of
such a complicated function, but the result agrees with our intuition. If you
throw a die, hoping to throw a "2", then the probability is 1/6. If
you throw it 6 times, you would expect to get one throw with value
"2". The mean or expected value for 6 throws is (1/6)(6) = 1. For
such a simple expression, the proof that it is in fact the mean is rather
involved. From the definition of the mean using a distribution function, the
binomial mean is: 

The goal is to reduce this expression to just np. Since the first term in the sum
is zero, since x=0, we can replace the sum with a sum starting from 1. 

Now cancel the common factor of x
appearing in numerator and denominator 

Since the summation index is a dummy variable, we make
the change of variables x' = x  1. 

Now factor out np. 

The terms in the summation above are just the binomial function
for n1 trials, and you are summing it over all values of x  so that sum
must be just 1. The expression then reduces to the desired expression for the
mean.

Mean
of the
binomial distribution. 
Since the Gaussian
and Poisson distributions are approximations to the binomial distribution,
this expression for the mean applies to them as well. 
If you have a collection
of n distinguishable objects, then the number of ways you can pick a number r
of them (r < n) is given by the permutation relationship: 

For example if you have six persons for tennis, then the
number of pairings for singles tennis is: 

But this really double counts, because it treats the
a:b match as distinct from the b:a match for players a and b. So in only 15
matches you could produce all distinguishable pairings. If you don't want to
take into account the different permutations of the elements, then you must
divide the above expression by the number of permutations of r. This result
is called a "combination". The combination relationship is: 

The number of tennis matches is then the combination: 

The permutation
relationship gives you the number of ways you can choose r objects or events
out of a collection of n objects or events. 

As in all of basic probability, the relationships come
from counting the number of ways specific things can happen, and comparing
that number to the total number of possibilities. If you are making choices
from n objects, then on your first pick you have n choices. On your second
pick, you have n1 choices, n2 for your third choice and so forth. As
illustrated before for 5 objects, the number of ways to pick from 5 objects
is 5!.


The number of permutations of r objects out of n is
sometimes what you need, but it has the drawback of overcounting if you are
interested in the number of ways to get distinguishable collections of
objects or events. For the purposes of card playing, the following ways of
drawing 3 cards are equivalent: 

So the permutation relationship overcounts the number
of ways to choose this combination if you don't want to make a distinction
between them based on the order in which they were chosen. The factor of
overcounting in this case is 6 = 3!, the number of permutations of 3 objects.
The number of distinguishable collections of r objects chosen from n is
obtained by dividing the permutation relationship by r! . It is usually
written 

For a group of n objects or events which are broken up
into k subsets, the above relationship is generalizable to give the number of
distinguishable permutations of the n objects. 

Here the terms in the denominator are
the populations of the k subsets. An application of this relationship is in
determining the number of possible energy states in an energy distribution
for distinguishable particles. 
The probability for a given
event can be thought of as the ratio of the number of ways that event can
happen divided by the number of ways that any possible outcome could happen.
If we identify the set of all possible outcomes as the "sample
space" and denote it by S, and label the desired event as E, then the
probability for event E can be written 

In the probability of a throw of a pair of dice, bet on
the number 7 since it is the most probable. There are six ways to throw a 7,
out of 36 possible outcomes for a throw. The probability is then 

The idea of an "event" is a very general one.
Suppose you draw five cards from a standard deck of 52 playing cards, and you
want to calculate the probability that all five cards are hearts. This
desired event brings in the idea of a combination. The number of ways you can
pick five hearts, without regard to which hearts or which order, is given by
the combination 

while the total number of possible outcomes is given by
the much larger combination 

The same basic probability expression is used, but it
takes the form


So drawing a fivecard hand of a single selected suit
is a rare event with a probability of about one in 2000. If you want the probabability that any one of a number of disjoint events
will occur, the probabilities of the single events can be added. For example,
the probability of drawing five cards of any one suit is the sum of four
equal probabilities, and four times as likely. In boolean language, if the
events are related by a logical OR, then the probabilities add. If the events are related by a logical AND, the resultant probability is
the product of the individual probabilities. If you want the probability of
throwing a 7 with a pair of dice AND throwing another 7 on the second throw,
then the probability would be the product 

The expression for probability must
be such that the addition of the probabilities for all events must be 1. Constraining
the sum of all the probabilities to be 1 is called "normalization".
When you calculate the probability by direct counting processes like those
discussed above, then the probabilities are always normalized. But when you
develop expressions for the probability of events in nature, you must make
sure that your probability expression is normalized. 
Standard Deviation for Particle Position
A free particle which is
constrained to be between x=0 and x=L has a distribution function which is just
a constant. The relationship for the standard deviation of the position is
the square root of the integral 

Normalizing the distribution gives the value for C. 

The mean value of x is 

Using this, the standard deviation becomes 

The average in the square root is 

The resulting standard deviation for the free particle
is


Stirling's Approximation for n!
When evaluating distribution functions for statistics,
it is often necessary to evaluate the factorials of sizable numbers, as in
the binomial distribution: 

A helpful and commonly used approximate relationship for the evaluation
of the factorials of large numbers is Stirling's approximation: 

A slightly more accurate approximation is the following 

but in most cases the difference is small. This additional term does give
a way to assess whether the approximation has a large error. Stirling's
approximation is also useful for approximating the log of a factorial, which
finds application in evaluation of entropy in terms of multiplicity, as in
the Einstein solid. The log of n! Is but the last term
may usually be neglected so that a working approximation is: 
