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.

Spherical Bessel Functions

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:


Properties of the Derivative

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 Non-homogeneous Differential Equation

An example of a first order linear non-homogeneous differential equation is

Having a non-zero value for the constant c is what makes this equation non-homogeneous, 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 non-homogeneous 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 non-homogeneous 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.

Binomial Distribution

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(n-1)(n-2)...(n-r+1) = n!/(n-r)! 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!(n-r)!] which is called the combination.

The probability of getting a certain outcome r times is pr, but if you do that AND get some other outcome for the remainder of the tries (associated probability (1-p)n-r) 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.

Functions for Statistics

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.

The Gamma 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.

Indefinite Integrals of Exponents and Logs

Indefinite Integrals of Trigonometric Functions

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

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.

Legendre Polynomials

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 Pn(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:


Associated Legendre Functions

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

where Pn(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.

Natural Log Series

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 n-1 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.

Permutations and Combinations

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 n-1 choices, n-2 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.


Basic Probability

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 five-card 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: