Unit 6: Introduction to Fourier series - Subjective Questions

Definition: A Fourier series represents a periodic function as a sum of constant, sine, and cosine terms. For a function of period , it is written as

• Constant term: is the mean value of the function over one period.
• Fundamental harmonic: The terms have the same period as the function.
• Higher harmonics: Terms involving and for have frequencies that are integral multiples of the fundamental frequency.
• The symbol indicates Fourier representation; at a discontinuity, the series may converge to the mean of the one-sided limits rather than to itself.

Assume that

Using the orthogonality relations on :

and

Integrating the series gives

Multiplying by and integrating gives

Similarly, multiplying by gives

These are called Euler's formulae for the Fourier coefficients.

For a function defined on and extended periodically with period , its Fourier series is

where

The trigonometric terms are orthogonal over , which permits each coefficient to be determined independently.

A periodic function has a convergent Fourier expansion under the usual Dirichlet conditions if, over any one period:

• is absolutely integrable, so is finite.
• has only a finite number of maxima and minima.
• has only a finite number of finite discontinuities.
• The function is single-valued and piecewise continuous.

Under these conditions:

• At a point where is continuous, the Fourier series converges to .
• At a jump discontinuity , it converges to

These conditions are sufficient, although they are not necessary in every advanced formulation of Fourier theory.

Suppose satisfies the Dirichlet conditions and has a jump discontinuity at . The Fourier series converges there to the average of the one-sided limits:

It does not generally converge to the assigned value .

At an endpoint of an interval, the periodic extension must be considered. For a function defined on ,

Near a jump, partial sums may overshoot and oscillate. This behavior is called the Gibbs phenomenon. Increasing the number of terms narrows the oscillatory region, but the maximum relative overshoot does not disappear completely.

Let the interval length be . Introduce the change of variable

As varies from to , varies from to . The Fourier expansion becomes

Since , the coefficients are

Thus, a change of interval converts the standard -periodic formula into a formula appropriate for any finite interval.

Since is odd, . Therefore,

Using integration by parts,

Hence,

Therefore,

At , the periodic extension has a jump from to , so the Fourier series converges to .

A function is even if

For an even function, is odd. Its integral over is zero, so

The remaining coefficients are

Thus, the Fourier series contains only cosine terms:

This is called a Fourier cosine series.

A function is odd if

For an odd function, both and are odd. Therefore,

The sine coefficients are

Hence, the Fourier series contains only sine terms:

This is called a Fourier sine series.

• General function: It normally has a constant term, cosine terms, and sine terms:

• Even function: Since , all sine coefficients vanish. Its expansion contains only the constant and cosine terms.
• Odd function: Since , the constant and cosine coefficients vanish. Its expansion contains only sine terms.
• Practical importance: Identifying symmetry before integration reduces the number of coefficient calculations and often changes integrals over into twice the corresponding integrals over .

To obtain a half-range cosine series, extend evenly to by defining

Because is even, all sine coefficients vanish. Its Fourier series is

where

and

The resulting periodic extension has period and is symmetric about the vertical axis.

To obtain a half-range sine series, extend oddly to by defining

Because is odd, . Therefore,

where

The odd extension and its periodic continuation have period . At and , every sine term is zero, so the sine series converges to zero there whenever interpreted directly at those endpoints.

For the half-range cosine series,

First,

so . Also,

Thus for even and for odd . Therefore,

for .

The half-range sine series has the form

The coefficients are

Integration by parts gives

Hence,

Therefore,

for .

Since is even, . The constant coefficient is

For ,

Integrating twice by parts gives

and hence

Therefore,

or

The function is odd, so . Its sine coefficients are

Therefore,

The Fourier series is

At and , the periodic extension has jumps between and . Consequently, the Fourier series converges to

at these points.

The coefficients are

and

Thus for odd and zero for even . Hence,

At , the one-sided limits are and , so the series converges to . The periodic extension also has jumps at , and the series converges to there as well.

Since is even, all sine coefficients vanish:

The constant coefficient is

For ,

Thus for even , while

for odd . Therefore,

Only odd-indexed cosine terms occur.

Half-range cosine series:

• It is obtained using an even extension of from to .
• It contains a constant term and cosine terms.
• Its coefficients are

Half-range sine series:

• It is obtained using an odd extension of .
• It contains only sine terms.
• Its coefficients are

At interior points of , both series represent under the Dirichlet conditions. At endpoints, the represented values are determined by the relevant periodic extension. A sine series is zero at and , whereas a cosine series can represent nonzero endpoint values when its even periodic extension is continuous there.

Euler's identity is

For a function of period , the complex Fourier series is

where

The trigonometric form is

For , the coefficients are related by

Conversely,

For a real-valued function, .