Dirichlet conditions. Sectionally continuous (or piecewise continuous) function. Fourier series. Even and odd functions. Half range series. Rate of convergence. Harmonic. Complex Fourier series. Parseval�s identity. Wave symmetry. Half-wave, Quarter-wave, and Hidden Symmetries. Orthogonal properties of sine and cosine functions.

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Dirichlet conditions. Sectionally continuous (or piecewise continuous) function. Fourier series. Even and odd functions. Half range series. Rate of convergence. Harmonic. Complex Fourier series. Parseval’s identity. Wave symmetry. Half-wave, Quarter-wave, and Hidden Symmetries. Orthogonal properties of sine and cosine functions.

Dirichlet conditions. The following conditions on a function defined over some interval [a, b] are called the Dirichlet conditions:

(a) it is continuous except for a finite number of discontinuities

(b) it has only a finite number of maxima and minima.

Def. Sectionally continuous (or piecewise continuous) function. A function f (x) is said to be sectionally continuous (or piecewise continuous) on an interval a x b if the interval can be subdivided into a finite number of intervals in each of which the function is continuous and has finite right and left hand limits. See Fig. 1. The requirement that a function be sectionally continuous on some interval [a, b] is equivalent to the requirement that it meet the Dirichlet conditions on the interval.

Fourier series. Let f (x) be a sectionally continuous function defined on an interval c < x < c + 2L. It can then be represented by the Fourier series

where

Derivation of formulas

At a point of discontinuity f (x) is given a value equal to its mean value at the discontinuity

i.e. if x = a is a point of discontinuity, f (x) is given the value

Note. The above Dirichlet conditions (a) and (b) are sufficient, but not necessary, conditions for the convergence of the series.

The Fourier series representation of f (x) is a periodic function with period 2L. It represents the function f (x) in the interval c < x < c + 2L and then infinitely repeats itself along the x-axis (in both positive and negative directions) outside the interval such that for any x, f (x + 2L) = f (x). See Fig. 2. Because of this, a Fourier series can be used to represent either a function f (x) defined only on an interval c < x < c + 2L or a periodically repeating function such as that shown in Fig. 2. If we wish to represent a periodic function such as that of Fig. 2 we simply find the Fourier representation for one of its cycles.

Def. Even function. A function whose value does not change when the sign of the independent variable is changed. That is, a function such that f (-x) = f (x).

Examples. x², 3x⁴ - 2x² + 1, cos x, e^x + e^-x

See Fig. 3a. An even function is symmetric with respect to the y axis.

Def. Odd function. A function whose sign changes, but whose absolute value does not change, when the sign of the dependent variable is changed. That is, a function such that f (-x) = -f (x).

Examples. x³, x⁵ -3x³ + 5x, sin x

See Fig. 3b. An odd function is symmetric with respect to the origin.

Fig. 3c shows a function that is neither even nor odd.

Products of even and odd functions. The product of two even or of two odd functions is an even function. The product of an even and an odd function is odd.

In the Fourier series corresponding to an even function only cosine terms (and possibly a constant) can be present. In the Fourier series corresponding to an odd function only sine terms can be present.

Half range Fourier cosine series. If f (x) is an even function, the coefficients in the Fourier series of f (x) are given by the formulas

This is called the half range Fourier cosine series.

Proof

Half range Fourier sine series. If f (x) is an odd function, the coefficients in the Fourier series of f (x) are given by the formulas

This is called the half range Fourier sine series.

Let us observe that the evenness and oddness of a function are not intrinsic properties of a graph but depend upon its relation to the vertical axis of the coordinate system. If in Fig. 4 the line AA' is chosen as the vertical axis, the graph defines an even function whose Fourier expansion will contain only cosine terms. If, on the other hand, line BB' is chosen as the vertical axis, the graph defines an odd function whose Fourier expansion will contain only sine terms. If a general line such as CC' is chosen, the graph defines a function that is neither even nor odd and both sines and cosines will appear in its Fourier series.

Often a half range series represents some function that has been defined in the interval (0, L) — i.e. half of the interval (-L, L) accounting for the term half range — and then specified as either even or odd, thus implicitly defining it in the other half of the interval. Half range series are important in certain problems in which expansions containing only sine terms or expansions containing only cosine terms must be constructed. It may be that the conditions of a problem require us to represent a function defined on the interval (0, L) by either a sine series or a cosine series, the values of the function outside this interval being of no interest to us. We are thus left free to define the function outside the interval (0, L) in any way we wish. We can then assume the function to be either even or odd, thus giving it either a cosine or sine representation.

Example 1. Consider the function

f (x) = x - x² -1 < x < 1

shown in Fig. 5. Its Fourier expansion for this interval (-1, 1) is given by

where the coefficients a_n and b_n are given by the formulas

The graph of its Fourier expansion 6) is shown in Fig. 6, a periodically repeating graph. Suppose now we are only really interested in that portion of the function in the interval (0, 1) and we wish to represent that portion with a cosine series. We can do that. We simply use the half range Fourier cosine series, computing its coefficients from formulas 3) above. The a_n coefficients, as computed using 3), are

and the series representation is

See Fig. 7. The graph is symmetric about the vertical axis.

Suppose, on the other hand, we wish to represent the portion of the function in the interval (0, 1) with a sine series. We can also do that. We simply use the half range Fourier sine series computing its coefficients from formulas 4) above. The formula for the b_n coefficients, as computed using 4), is

and the series representation is

Its graph is shown in Fig. 8. The graph is symmetric about the origin.

Rate of convergence of a Fourier series. It is of value to consider the way the partial sums of a Fourier series converge to the function being represented. Consider the function defined by

Letting c = -π, L = π, in formulas 2) above we obtain the Fourier series expansion

Derivation of series

Let s_n(x) be the sum of the first n terms of the series. Then

In Fig. 9 one can see how the partial sums come closer and closer to the given function. For n = 4, 5, 6, .... the partial sums are almost indistinguishable from f (x).

Theorem 1. As n becomes infinite, the coefficients in the Fourier series of a function that satisfies the Dirichlet conditions always approach zero at least as rapidly as c/n, where c is a constant independent of n. If the function has one or more points of discontinuity, the coefficients can decrease no faster than this. In general, if a function and its various derivatives all satisfy the Dirichlet conditions, and if the k-th derivative is the first one which is not everywhere continuous, then the coefficients in the Fourier series of the function approach zero as c/n^k+1as n becomes infinite.

In more concise but less exact language, the theorem asserts that the smoother the function, the faster its Fourier series converges.

Example 2. Consider the functions of Figs. 6, 7 and 8. The graph in Fig. 6 is discontinuous and we note from formulas 7) that the coefficients of the sine terms of its series decrease only at a rate proportional to 1/n. The graph of Fig. 7 is continuous but it has sharp corners where the tangent makes abrupt changes in direction. The coefficients of its series decrease at a rate proportional to 1/n² as seen by inspecting 8). The graph of Fig. 8 is not only continuous but is also smooth i.e. there are no points at which the tangent changes direction abruptly. We note from formula 10) that its series decrease at a rate proportional to 1/n³.

Theorem 2. The integral of any function which satisfies the Dirichlet conditions can be found by termwise integration of the Fourier series of the function.

Theorem 3. Let f (x) be a continuous function satisfying the Dirichlet conditions. If f '(x) also satisfies the Dirichlet conditions, then wherever it exists, f '(x) can be found by termwise differentiation of the Fourier series of f (x) .

Termwise differentiation of a Fourier series multiplies the coefficients a_n and b_n by ± nπ/L and thus tends to slow down convergence and may result in divergence. In termwise integration, on the other hand, the coefficients a_n and b_n are divided by ±nπ/L resulting in a series in which convergence is enhanced.

Fourier series as a sum of harmonics. Let f (t) be a sectionally continuous function that is defined over an interval c < t < c + P. If we let L = P/2 and ω₀= 2π/P = π/L in formulas 1) and 2) above we get the following representation for a Fourier series:

14) f (t) = a₀/2 + a₁ cos ω₀t + a₂ cos 2ω₀t + .... + a_n cos nω₀t + ....

+ b₁ sin ω₀t + b₂ sin 2ω₀t + .... + b_n sin nω₀t + ....

where

This series can also be written in the form

where

or in the form

where

Proof

The quantity ω_n = nω₀ is called the n-th harmonic of the series. The first harmonic is commonly called the fundamental component since it has the same period as the function. The coefficients A_n are known as the harmonic amplitudes and the angles θ_n and are called phase angles.

We thus see from 16) and 17) that a function defined on some interval p can be viewed as being composed of the sum of either sinusoidal or cosinusoidal waveforms i.e. composed of the sum of its various sinusoidal or cosinusoidal harmonics.

We shall now present a model that will provide insight into the meaning of the quantity ω. The function y = A cos ωt can be viewed as a graph of the displacement of a vibrating particle in simple harmonic motion as a function of time t.

Let Q be a point moving with constant speed around the circumference of a circle of unit radius. See Fig. 10. Let AB be the horizontal diameter of the circle, θ be the angle BOQ, and P be the projection of Q on AB as shown in the figure. As Q moves around the circle with uniform speed, point P moves back and forth between B and A with what is termed “simple harmonic motion” (the motion described by a particle in a freely vibrating string). Let θ₀ be the angular position of Q at time t = 0, θ be its angular position at time t and ω be its angular velocity in radians / sec. Then the angular position θ of Q at any time t is given by ωt + θ₀. Since = cos θ and = 1, at any time t = cos θ = cos (ωt + θ₀). Hence the motion described by point P is given by the function y = cos (ωt + θ₀). If θ₀ = 0 the function becomes y = cos ωt. If the radius of the circle is A instead of unity, then the function becomes y = A cos (ωt + θ₀).

Thus we see that the intuitive interpretation of ω is angular velocity in radians / sec. — in connection with this model. The angular velocity of point Q may be given in other forms. It may be given as the number of revolutions made by Q about the circle per second. This is called its frequency f. The frequency f and angular velocity ω are related by

ω = 2πf. The frequency f can also be stated as the number of vibrations per sec. of the point P, which is the same as the number of revolutions of Q per sec. The angular velocity of Q can also be specified by giving the period p, which is the time required for one revolution of Q about the circle, or one vibration of P. The period p and frequency f are related by f = 1/p.

We can obtain a similar interpretation for the function y = sin (ωt + θ₀) if we let P be the projection of Q on the vertical diameter CD of the circle as shown in Fig. 11.

Fig. 12 shows a graph of the general function

for n = 1 and n = 3. For n = 1 the interval (0, P) contains one cycle of the sine function and for n = 3 it contains three cycles. For the general case n, it contains n cycles.

Fig. 13 shows the graph of the general equation y = sin ax. The period p of this function is given by p = 2π/a. Thus the function y = sin 10x has a period of p = 2π/10 = π/5. The period corresponds to the range of values of x corresponding to a complete cycle of the function. Thus it is found by solving the equation ax = 2π for x giving x = 2π/a.

If the independent variable in a sine or cosine function is not time, the above interpretation for ω and f break down. There remains, however, the concept of the period p.

Complex Fourier series. If we substitute the Euler formulas

into the sine and cosine terms of 1) above we obtain the Complex Fourier Series

where

or, equivalently,

where f (x) is defined on the interval c < x < c + 2L. In most cases c will be either - L or 0.

Proof

Parseval’s identity for Fourier series. Given the Fourier series 1) above for a function f(x). Parseval’s identity states that

An important consequence of this theorem is Riemann’s theorem:

Wave symmetry. Half-wave, Quarter-wave, and Hidden Symmetries

Even and odd functions. Even and odd functions are examples of functions possessing types of symmetry. An even function is symmetric with respect to the y-axis and an odd function

is symmetric with respect to the origin.

Half-wave Symmetry. Let f(t) be a periodic function with period T. Then f(t) is said to have half-wave symmetry if it satisfies the condition

See Fig. 14. Note that the negative portion of the wave in the figure is the mirror image of the positive portion displaced horizontally by a half period.

Theorem 4. If a periodic function f(t) is half-wave symmetric, then

Quarter-wave symmetry

Even quarter-wave symmetry. If a function possessing half-wave symmetry is also even, it is said to possess even quarter-wave symmetry. See Fig. 15 (a).

Odd quarter-wave symmetry. If a function possessing half-wave symmetry is also odd, it is said to possess odd quarter-wave symmetry. See Fig. 15 (b).

Hidden symmetry. Often the symmetry of a periodic function is obscured by a constant term. For example, consider the function shown in Fig. 16(a). This function possesses a hidden symmetry obscured by the constant term. If we construct a new function from f(t) by subtracting a constant A/2 from f(t), (which has the effect of shifting the horizontal axis up by an amount of A/2), we get the function shown in Fig. 16(b) which is an odd function.

Fourier series representation of periodic functions with symmetrical waveforms

Let f(t) be a periodic function with period T which satisfies the Dirichlet conditions. Then its Fourier series representation is given by

Theorem 5. If f(t + T)= f(t), then for any a

Sample problem: Find the Fourier series representation of a function

1] Fourier series representation of an even function. If f(t) is a periodic even function with period T the Fourier series consists of a constant and cosine terms only i.e.

and

2] Fourier series representation of an odd function. If f(t) is a periodic odd function with period T the Fourier series consists of sine terms only i.e.

and

3] Fourier series representation of a function with half-wave symmetry. If f(t) is a periodic function of period T with half-wave symmetry the Fourier series contains only odd harmonics i.e.

a_n = 0 for n even

b_n = 0 for n even

4] Fourier series representation of a function with even quarter-wave symmetry. If f(t) is a periodic function of period T with even quarter-wave symmetry the Fourier series consists of odd harmonics of cosine terms only i.e.

5] Fourier series representation of a function with odd quarter-wave symmetry. If f(t) is a periodic function of period T with odd quarter-wave symmetry the Fourier series consists of odd harmonics of sine terms only i.e.

Properties of Sine and Cosine functions

Def. Orthogonal functions. Two functions f₁(x) and f₂(x) are said to be orthogonal on a range (a, b) if

Def. Orthogonal system. A system of functions

is called orthogonal on the range (a, b) if every function is orthogonal to every other function i.e. if

Orthogonal properties of sine and cosine functions

The sequence of trigonometric functions

1, cos x, sin x, cos 2x, sin 2x, .... , cos nx, sin nx, .....

on the interval (-π, π) is one of the first and most important examples of orthogonal systems, appearing in works of Euler, D. Bernoulli, and d’Alembert in connection with problems on oscillation of strings. The study of it has played an essential role in the development of analysis.

Properties of sine and cosine

References

Wylie. Advanced Engineering Mathematics. Chap. 7

Spiegal. Laplace Transforms. (Schaum) Chap. 6

James & James. Mathematics Dictionary.

Taylor. Advanced Calculus.

Hsu. Fourier Analysis.