Unit 4: Fourier Series

MTH174 — Engineering Mathematics 4 min read

I. Orientation

Fourier series represent a periodic function as an infinite sum of sine and cosine harmonics. For a function with period (2L), the representation is built from the fundamental angular frequency (\pi/L) and its integer multiples. This provides a systematic method for analysing waveforms, vibrations, heat distributions, and other periodic phenomena.

A. Governing framework

Periodic function: A function (f(x)) has period (2L) when
[
f(x+2L)=f(x)
]
for every (x) where the function is defined.
Fourier representation: On ((-L,L)), the standard expansion is
[
f(x)\sim \frac{a0}{2}+\sum{n=1}^{\infty}
\left[a_n\cos\left(\frac{n\pi x}{L}\right)
+b_n\sin\left(\frac{n\pi x}{L}\right)\right].
]
Coefficient symbols: (a_0), (a_n), and (b_n) are Fourier coefficients; (n) is a positive integer; (L) is half the period.
Orthogonality principle: Sines and cosines of different harmonic orders are orthogonal over ((-L,L)), allowing each coefficient to be isolated by integration.
Convergence interpretation: At a point where (f) is continuous, the series converges to (f(x)); at a jump discontinuity, it converges to the average of the two one-sided limits.

II. Fourier Series — Introduction and Euler’s Formulae

A. Introduction and Euler's formulae

A Fourier series decomposes a periodic function into a constant component and harmonic sine and cosine components. Euler’s formulae determine the coefficients by exploiting orthogonality.

General coefficient formulae:
TEXT
```
  a0 = (1/L) ∫[-L,L] f(x) dx

  an = (1/L) ∫[-L,L] f(x) cos(nπx/L) dx,   n ≥ 1

  bn = (1/L) ∫[-L,L] f(x) sin(nπx/L) dx,   n ≥ 1
```
Here, (a_0) gives twice the mean value, while (a_n) and (b_n) measure the cosine and sine content of harmonic (n).
Orthogonality relations:
[
\int{-L}^{L}\cos\frac{m\pi x}{L}\cos\frac{n\pi x}{L}\,dx=0
\quad (m\ne n),
]
[
\int{-L}^{L}\sin\frac{m\pi x}{L}\sin\frac{n\pi x}{L}\,dx=0
\quad (m\ne n).
]
These relations eliminate all unwanted harmonics when a Fourier series is multiplied by a selected sine or cosine and integrated.
Complex Euler form: Euler’s identity
[
e^{i\theta}=\cos\theta+i\sin\theta
]
combines the two trigonometric components into complex exponentials. With (\omega0=\pi/L),
[
f(x)\sim\sum{n=-\infty}^{\infty}c_ne^{in\omega_0x},
\qquad
cn=\frac{1}{2L}\int{-L}^{L}f(x)e^{-in\omega_0x}\,dx.
]
The coefficient (c_n) is complex, and (i^2=-1).
Meaning of the constant term: The quantity (a_0/2) is the average value of (f) over one period:
[
\frac{a0}{2}=\frac{1}{2L}\int{-L}^{L}f(x)\,dx.
]

B. Fourier expansion procedure

The coefficients should be calculated before attempting to simplify the resulting series.

Step 1 — Identify the interval: Write the period as (2L), so the basis functions are (\cos(n\pi x/L)) and (\sin(n\pi x/L)).
Step 2 — Calculate (a_0), (a_n), and (b_n): Substitute the given (f(x)) into the integral formulae.
Step 3 — Simplify using symmetry: If (f) is even or odd, half the integrations vanish automatically.
Step 4 — State endpoint interpretation: At a discontinuity, use the midpoint of the left and right limits rather than the value assigned at the point.

III. Convergence and Conditions

A. Conditions for a Fourier expansion

A piecewise smooth function has a Fourier expansion under standard Dirichlet-type conditions. These conditions ensure that the required integrals exist and that the series has a meaningful pointwise limit.

Integrability: (f(x)) should be absolutely integrable over one period:
[
\int_{-L}^{L}|f(x)|\,dx<\infty.
]
This guarantees that the Fourier coefficients are finite.
Finite discontinuities: In any period, (f) may have only finitely many discontinuities, and each discontinuity should be finite. A finite jump is acceptable; an unbounded singularity is not covered by the elementary theorem.
Finite extrema: The function should have only finitely many maxima and minima in a period. Piecewise continuously differentiable functions satisfy this condition.
Piecewise smoothness: (f) and usually (f') should be continuous on each subinterval separated by finitely many points. A piecewise polynomial or piecewise trigonometric function is a common example.

Convergence result:

TEXT

  Fourier series at x =
      f(x),                         if f is continuous at x
      [f(x+) + f(x−)] / 2,          if f has a jump at x

Here, (f(x+)) and (f(x-)) are the right-hand and left-hand limits.

B. Functions having points of discontinuity

At a discontinuity, the Fourier series does not generally reproduce the assigned function value. Instead, it reproduces the midpoint of the limiting values from both sides.

Jump discontinuity: If
[
f(x^-)=A,\qquad f(x^+)=B,
]
then the series converges at (x) to
[
\frac{A+B}{2}.
]
The value (f(x)) at the single point does not affect the Fourier coefficients.
Step-function illustration: Suppose a periodic function has values (1) immediately to the left of (x=0) and (3) immediately to the right. At (x=0), its Fourier series converges to
[
\frac{1+3}{2}=2,
]
even if the function is defined as (1) or (3) at that point.
Gibbs phenomenon: Near a jump of size (B-A), partial sums overshoot and oscillate. Increasing the number of terms narrows the oscillatory region but does not completely remove the limiting overshoot; its approximate maximum size is about (9\%) of the jump.
Endpoint caution: When an interval is used to define one period, its endpoints may represent the same periodic point. The one-sided limits must therefore be interpreted using the periodic extension.

IV. Interval Transformation

A. Change of interval

The Fourier basis depends on the length of the chosen interval. Changing the interval changes the fundamental frequency and therefore changes the coefficient formulae.

Interval ((-L,L)): For period (2L),
[
f(x)\sim\frac{a0}{2}+\sum{n=1}^{\infty}
\left[a_n\cos\left(\frac{n\pi x}{L}\right)
+b_n\sin\left(\frac{n\pi x}{L}\right)\right].
]
Interval ((-\pi,\pi)): Taking (L=\pi) gives the familiar form
[
f(x)\sim\frac{a0}{2}+\sum{n=1}^{\infty}
(a_n\cos nx+b_n\sin nx),
]
with
[
an=\frac{1}{\pi}\int{-\pi}^{\pi}f(x)\cos nx\,dx.
]
General interval ((a,a+2L)): The coefficients can be integrated directly over any interval of length (2L):
[
a_n=\frac{1}{L}\int_a^{a+2L}f(x)\cos\left(\frac{n\pi x}{L}\right)\,dx.
]
A shifted coordinate (t=x-a-L) often converts the interval to ((-L,L)).
Scaling principle: If the interval is ((-c,c)), then (L=c), and the (n)-th harmonic is (\cos(n\pi x/c)), not (\cos(nx)). Omitting this scaling gives incorrect frequencies and coefficients.
Worked example: On ((-\pi,\pi)), (f(x)=x) is odd. Thus (a_0=a_n=0), while
[
bn=\frac{1}{\pi}\int{-\pi}^{\pi}x\sin nx\,dx
=\frac{2}{\pi}\int_0^\pi x\sin nx\,dx
=\frac{2(-1)^{n+1}}{n}.
]
Therefore,
[
x\sim2\left(\sin x-\frac{\sin2x}{2}+\frac{\sin3x}{3}-\cdots\right).
]

V. Symmetry of Functions

A. Even and odd functions

Symmetry reduces the amount of integration required and predicts which type of Fourier terms can occur.

Even function: (f(-x)=f(x)). Products of two even functions are even, so
[
bn=\frac{1}{L}\int{-L}^{L}f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx=0.
]
Only the constant and cosine terms remain.
Odd function: (f(-x)=-f(x)). Since an odd function times a cosine is odd,
[
a_0=0,\qquad a_n=0.
]
Only sine terms remain.

Reduction to ((0,L)):