10.4 Even and Odd Functions

Parseval's equation is an important part of the theory of Fourier series.

There is a proof of convergence of a Fourier series under more restrictive conditions.

In the next problem, we show how it is possible to improve the convergence of a series.

It is possible to show that the series converges slowly.

There are two classes of functions for which the Euler-Fourier formulas can be simplified.

There are examples of odd functions.

It is noted that according to Eq.

Two even functions have the same sum and product.

The product of two odd functions is even.

The proof of all these assertions are easy to follow.

If either function disappears identically, these statements may need to be changed.

The properties are clear from the interpretation of an integral in terms of area under a curve and follow immediately from the definitions.

The proof is the same for odd functions.

Even and odd functions are important in applications of the Fourier series since they have special forms.

There is a phenomenon near the points of discontinuity.

There is a wave.

The form of the expansion will be dictated by the purpose for which it is needed.

If there is a choice as to the kind of Fourier series to be used, the selection can be based on the rapidity of convergence.
The triangular wave is easier to approximate than the sawtooth wave because it is a smooth function.

The derivative of an even function is odd, and the derivative of an odd function is even.

This relation was discovered by a mathematician.

There are three different types of physical phenomena associated with the basic partial differential equations of heat conduction, wave propagation, and potential theory.

They are important in many branches of physics.
From a mathematical point of view, they are significant.
The linear equations of second order are the most significant and varied of the partial differential equations.
The heat conduction equation, wave equation, and potential equation are prototypes of each of the other categories.
A study of these three equations yields a lot of information.

Several methods for solving partial differential equations have been developed over the last two centuries.

The method of separation of variables is the oldest systematic method and was used by D'Alembert, Daniel Bernoulli, and Euler.
It is a method of great importance and frequent use today and has been refined and generalized.
To show how the method of separation of variables works, we need to look at a basic problem.
Modern scientists are interested in the mathematical study of heat conduction, which began in the 1800's.
The analysis of the transfer of heat away from its sources in high-speed machinery is an important technological problem.

There is a heat conduction problem for a straight bar of uniform cross section.

The sides of the bar are insulated so that they don't get hot.

Joseph Fourier was the prefect of the department of Isere (Grenoble) from 1802 to 1802 and was responsible for the first important investigation of heat conduction.

Basic papers on the subject were presented to the Academy of Sciences of Paris.
The papers were not published because they were criticized by the referees for lack of rigor.

A partial differential equation governs the variation of temperature in the bar.

The more general problem can be reduced to this special case.

The problem was described.

We could approach the problem by seeking solutions of the differential equation and boundary conditions, and then superposing them to satisfy the initial condition.

The plan can be implemented in this section.

Our goal is to find solutions to the differential equation and boundary conditions.

To find the needed solutions, we need to make a basic assumption about the form of the solutions that have far-reaching and unforeseen consequences.

Substituting from Eq.

It is necessary that both sides of the equation.

The partial differential equation was replaced by the two ordinary differential equations.

We are only interested in the solutions of Eq.

We now want to think about it.

The problem that we discussed in detail at the end of Section 10 is the same problem that we are talking about here.

The results obtained earlier should be referred to.

Now it's time to go to Eq.

There are solutions of Eqs.

We have often solved initial value problems by forming linear combinations of a set of fundamental solutions and choosing coefficients to satisfy the initial conditions.

The same step as in the present problem is to form a linear combination of functions and then choose the coefficients to satisfy Eq.
There are infinitely many problems and a general linear combination of them is an infinite series.

The infinite series of Eq.

will be assumed.

It is possible to get accurate results by only using a few terms of the series.

The units of cm2/sec are 2.

For convenience, this corresponds to a rod of a material that has thermal properties similar to copper and aluminum.
As heat is lost through the endpoints, the temperature decreases.

The warmest point in the bar is always the center because of the symmetry of the initial temperature distribution.

Find the equations if it's true.

If you put all the graphs on the same axes, you can see how the temperature distribution changes with time.

Let the rod be heated to a temperature of 100*C.