Unit 4: Fourier Series - Subjective Questions

Fourier series: A Fourier series represents a periodic function as an infinite sum of sine and cosine functions.

For a function with period , its Fourier series is

The coefficients are determined by

Purpose: Fourier series are used to express complicated periodic functions in terms of elementary trigonometric functions, which is useful in heat conduction, wave motion, vibrations, and electrical engineering.

Let the Fourier series of over be

Using the orthogonality relations of sine and cosine functions:

• for .
• .
• .
• .

Multiplying the series by and integrating gives

Multiplying by and integrating gives

Multiplying by and integrating gives

These are called Euler's formulae.

The trigonometric functions form an orthogonal system on the interval . The important relations are:

For equal indices,

and

The constant function is also orthogonal to the nonconstant sine and cosine terms. These properties allow all unwanted terms to vanish when the Fourier series is multiplied by a selected sine or cosine function and integrated. Consequently, one coefficient can be isolated at a time.

The usual Dirichlet conditions ensure that a Fourier series converges suitably. A function should satisfy the following conditions over one period:

• Single-valuedness: must be finite and single-valued.
• Periodic nature: The function must be periodic or defined on a finite interval for periodic extension.
• 有限 number of discontinuities: may have only a finite number of finite discontinuities.
• 有限 number of maxima and minima: The function should have only a finite number of maxima and minima in one period.
• Absolute integrability: The integral over one period must be finite.

At every point where is continuous, the Fourier series converges to . At a point of finite jump discontinuity , it converges to the average of the limiting values:

Suppose satisfies the Dirichlet conditions and has Fourier series .

• At a point where is continuous,

• At a jump discontinuity, let

Then the Fourier series converges to

• If the function is continuous at an endpoint after periodic extension, the series converges to the common endpoint value.

Thus, a Fourier series reproduces the function at continuous points but gives the midpoint of the jump at a finite discontinuity. Near a jump, oscillations known as the Gibbs phenomenon may occur. These oscillations become concentrated near the discontinuity as more terms are included, although their maximum overshoot does not completely disappear.

For a function defined on , take in the general Fourier series. Therefore,

The coefficients are obtained from Euler's formulae:

Hence, the required Fourier expansion is

The variable is used in the integrals to distinguish it from the independent variable .

On the interval , the fundamental trigonometric functions are and . When the interval is changed to , the fundamental angular frequency becomes .

Therefore, the Fourier series becomes

where

The resulting series has period . The change in interval therefore changes the wavelengths of the basis functions but not the general method of calculating the coefficients.

Let

so that

The interval in the variable corresponds to in the variable . If the standard Fourier series is

then substituting gives

Using , the coefficients become

Thus, a change of variable transforms the standard Fourier series into the Fourier series for any symmetric interval .

A function is even if

A function is odd if

Their importance follows from the parity of trigonometric functions:

• is even.
• is odd.

If is even, then is odd, so

Consequently, the Fourier series contains only cosine terms:

If is odd, then is odd and . Hence, the Fourier series contains only sine terms:

This property greatly reduces the amount of integration required.

Let be even. Since is also even, their product is even. Therefore,

Similarly,

Since is even and is odd, the product is odd. The integral of an odd function over is zero, so

Hence, the Fourier series of an even function is

with

Let be odd. Since is odd and is even, the product is odd. Therefore,

and

Since both and are odd, their product is even. Thus,

Therefore, the Fourier series of an odd function is

where

A half-range Fourier series represents a function specified only on by extending it to the interval .

There are two standard extensions:

• Even extension: Define . The resulting Fourier series contains only cosine terms and is called the half-range cosine series:

where

• Odd extension: Define . The resulting Fourier series contains only sine terms and is called the half-range sine series:

where

To obtain a half-range cosine series, extend from to as an even function:

For the even extension, all sine coefficients vanish, so the Fourier series is

Using the even-function formulas,

and

Therefore, the half-range cosine expansion is

This series is particularly useful when the boundary conditions of a physical problem involve prescribed values of the function or zero derivative at the endpoints.

To obtain a half-range sine series, extend from to as an odd function:

For the odd extension, the constant and cosine coefficients vanish. Thus, the Fourier series contains only sine terms:

The coefficients are

Hence, the half-range sine expansion is

This series is especially useful for problems in which the function is zero at an endpoint or satisfies a boundary condition naturally represented by sine functions.

The main differences are as follows:

For a full-range series,

For a half-range cosine series, only cosine terms occur, while for a half-range sine series, only sine terms occur. The choice depends on the desired extension and the boundary conditions of the problem.

Let be piecewise continuous with a finite jump discontinuity at . Suppose its one-sided limits are

The Fourier series does not generally converge to either one-sided value. Instead, it converges to their arithmetic mean:

At all points where is continuous,

If the function is defined with a value different from this average at the discontinuity, the Fourier series still converges to the average, not to the assigned value. Near the discontinuity, partial sums can overshoot and oscillate. This is the Gibbs phenomenon. The oscillations become narrower as the number of terms increases, but the relative overshoot remains near the jump.

The function is odd because

Therefore,

and only sine coefficients need to be calculated:

Since is even,

Integrating by parts,

Thus,

Hence,

The odd symmetry eliminates all cosine terms and reduces the calculation to one integral over .

For a half-range sine series on ,

where

Integrating by parts,

Therefore,

Since and ,

Consequently, the half-range sine series is

This is the sine series corresponding to the odd extension of to .

For the half-range cosine series on ,

where

Thus,

For ,

Using integration by parts,

Therefore,

Hence, for even and for odd . The series is

This corresponds to the even extension of .

Suppose a periodic function is piecewise continuous and has a jump at . Its Fourier series has the form

The coefficients are calculated by integrating over one complete period, even if the function is defined by different formulas on different subintervals. Thus, if

then, for example,

A similar expression applies to . At points of continuity, the series equals the function. At , it converges to

Therefore, the Fourier representation is valid pointwise with the midpoint interpretation at jumps.