Unit 6: Introduction to Fourier series

MTH165 — Mathematics For Engineers 9 min read

I. Orientation — Representing Periodic Functions

A Fourier series represents a periodic function as an infinite sum of constant, sine, and cosine terms. Developed from the work of Joseph Fourier (1768–1830), it converts a function in the time or spatial domain into components with distinct frequencies.

A. Defining Framework

Fourier analysis rests on periodicity, orthogonality, and the approximation of functions by trigonometric components.

Periodic function: A function (f(x)) has period (T>0) when
TEXT
```
  f(x + T) = f(x)
```
where (x) is the independent variable and (T) is the fundamental period when it is the smallest positive period.
Fundamental angular frequency: For a function of period (T),
TEXT
```
  ω₀ = 2π/T
```
where (\omega_0) is the fundamental angular frequency and (n\omega_0), for positive integer (n), is the (n)th harmonic.
General Fourier series:
TEXT
```
  f(x) ~ a₀/2 + Σ[n=1 to ∞]{aₙ cos(nω₀x) + bₙ sin(nω₀x)}
```
Here (a_0/2) is the mean component, (a_n) and (b_n) are Fourier coefficients, and the symbol (\sim) denotes representation or convergence in the Fourier-series sense.
Orthogonality: Over one complete period, distinct sine and cosine harmonics have zero inner product. For example, on ((-\pi,\pi)),
TEXT
```
  ∫[-π to π] cos(mx)cos(nx) dx = 0,   m ≠ n
  ∫[-π to π] sin(mx)sin(nx) dx = 0,   m ≠ n
  ∫[-π to π] sin(mx)cos(nx) dx = 0
```
where (m) and (n) are positive integers. Orthogonality allows each coefficient to be isolated by integration.
Engineering interpretation: The constant term gives the average value, while each sine–cosine pair describes a harmonic component. This supports spectral analysis of signals, vibrations, heat distributions, electrical waveforms, and periodic loading.

II. Fourier Representation — Coefficients and Euler’s Relations

A. Introduction and Euler's formulae

Euler’s formulae determine the coefficients of a Fourier series by projecting a periodic function onto mutually orthogonal trigonometric functions.

Standard interval: For a function of period (2\pi), defined over ((-\pi,\pi)),
TEXT
```
  f(x) ~ a₀/2 + Σ[n=1 to ∞]{aₙ cos(nx) + bₙ sin(nx)}
```
Euler’s coefficient formulae:
TEXT
```
  a₀ = (1/π)∫[-π to π] f(x) dx

  aₙ = (1/π)∫[-π to π] f(x)cos(nx) dx

  bₙ = (1/π)∫[-π to π] f(x)sin(nx) dx
```
Here (a_0) determines the average term, (a_n) measures the cosine content, (b_n) measures the sine content, and (n=1,2,3,\ldots).
Origin of the factor (a_0/2): Integrating the proposed series over ((-\pi,\pi)) eliminates every sine and cosine term:
TEXT
```
  ∫[-π to π] f(x) dx = πa₀
```
Therefore the constant term in the series is (a_0/2), equal to the mean value of (f) over one period.

General period (2L): If (f(x+2L)=f(x)), its Fourier series on ((-L,L)) is

TEXT

  f(x) ~ a₀/2 + Σ[n=1 to ∞]{
      aₙ cos(nπx/L) + bₙ sin(nπx/L)
  }

with

TEXT

  a₀ = (1/L)∫[-L to L] f(x) dx

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

  bₙ = (1/L)∫[-L to L] f(x)sin(nπx/L) dx

Here (2L) is the period and (L>0) is the half-period.

Complex form: Euler’s identity (e^{i\theta}=\cos\theta+i\sin\theta) gives
TEXT
```
  f(x) ~ Σ[n=-∞ to ∞] cₙe^(inω₀x)

  cₙ = (1/T)∫[x₀ to x₀+T] f(x)e^(-inω₀x) dx
```
where (i^2=-1), (c_n) is the complex Fourier coefficient, and (x_0) is any starting point for a complete period.
Coefficient relationship:
TEXT
```
  c₀ = a₀/2
  cₙ = (aₙ - ibₙ)/2
  c₋ₙ = (aₙ + ibₙ)/2
```
These relations show that the real trigonometric and complex exponential forms contain the same information.

III. Convergence — Admissible Functions and Discontinuities

A. Conditions for a Fourier expansion and functions having points of discontinuity

The usual sufficient conditions ensure that a piecewise regular periodic function has a Fourier series with predictable pointwise convergence.

Dirichlet conditions: Over any one period, it is sufficient that (f):
- is absolutely integrable, so (\int |f(x)|\,dx) is finite;
- has only finitely many finite discontinuities;
- has only finitely many maxima and minima;
- is single-valued and finite except possibly at isolated jump discontinuities.
Sufficient, not necessary: A function may possess a valid Fourier expansion without satisfying every classical Dirichlet condition. The conditions provide a practical engineering test rather than a complete characterization of Fourier convergence.
Continuous point: If (f) is continuous at (x=x_0), then its Fourier series converges there to
TEXT
```
  S(x₀) = f(x₀)
```
where (S(x_0)) denotes the sum of the Fourier series at (x_0).
Jump discontinuity: If the finite one-sided limits exist, the series converges to their arithmetic mean:
TEXT
```
  S(x₀) = {f(x₀⁻) + f(x₀⁺)}/2
```
Here (f(x_0^-)) and (f(x_0^+)) are the left-hand and right-hand limits. The assigned value (f(x_0)) itself does not affect the coefficients because changing finitely many point values does not alter the integrals.
Periodic endpoints: At the endpoints of a selected interval, the periodic extension must be considered. On ((-L,L)),
TEXT
```
  S(L) = S(-L) = {f(L⁻) + f(-L⁺)}/2
```
because (x=L) and (x=-L) represent the same point in the periodic continuation.
Gibbs phenomenon: Near a jump, finite partial sums develop oscillatory overshoots and undershoots. Increasing the number of terms narrows the oscillatory region but does not eliminate the limiting peak overshoot, which is approximately (9\%) of the jump magnitude.
Worked example: If the periodic extension has (f(0^-)=2) and (f(0^+)=8), its Fourier series converges at (x=0) to
TEXT
```
  S(0) = (2 + 8)/2 = 5
```
regardless of the separately assigned value of (f(0)).

IV. Rescaling the Domain — Fourier Series on General Intervals

A. Change of interval

A change of interval transforms a nonstandard domain into a standard Fourier interval while preserving the relative position of each point.

Interval ((-L,L)): Substitute
TEXT
```
  t = πx/L
```
where (x\in(-L,L)) and (t\in(-\pi,\pi)). Then the harmonics become (\cos(n\pi x/L)) and (\sin(n\pi x/L)).
Arbitrary interval ((a,b)): Its length is (T=b-a), so define
TEXT
```
  t = 2π(x-a)/(b-a)
```
where (a) and (b) are the interval endpoints. A complete Fourier representation then uses angular frequency (2\pi/(b-a)).
Direct coefficient form on ((a,a+T)):
TEXT
```
  f(x) ~ a₀/2 + Σ[n=1 to ∞]{
      aₙ cos(2πnx/T) + bₙ sin(2πnx/T)
  }
```
with coefficients obtained by multiplying the corresponding integrals over one period by (2/T). A phase-shifted basis using (x-a) may instead be used for simpler limits.
Scaling principle: Enlarging the period lowers the fundamental frequency:
TEXT
```
  ω₀ = 2π/T
```
Thus a function with period (8) has fundamental angular frequency (\pi/4), and its (n)th harmonic has angular frequency (n\pi/4).
Engineering significance: Changing the interval allows measurements taken over physical lengths, time windows, or angular ranges to be expanded without artificially replacing their units or period.

V. Symmetry — Simplifying Fourier Coefficients

A. Even and odd functions

Parity removes entire families of Fourier coefficients and often halves the required integration.

Even functions: A function is even when
TEXT
```
   f(-x) = f(x)
```
- Symmetry: Its graph is symmetric about the vertical axis.
- Integral property:
  TEXT
```
     ∫[-L to L] f(x) dx = 2∫[0 to L] f(x) dx
```
- Fourier consequence: Since an even function multiplied by (\sin(n\pi x/L)) is odd,
  TEXT
```
     bₙ = 0
```
  so the expansion contains only the constant and cosine terms.
- Reduced coefficients:
  TEXT
```
     a₀ = (2/L)∫[0 to L] f(x) dx

     aₙ = (2/L)∫[0 to L] f(x)cos(nπx/L) dx
```
Odd functions: A function is odd when
TEXT
```
   f(-x) = -f(x)
```
- Symmetry: Its graph has rotational symmetry through the origin.
- Integral property:
  TEXT
```
     ∫[-L to L] f(x) dx = 0
```
- Fourier consequence: Products with the constant function or any cosine harmonic are odd, so
  TEXT
```
     a₀ = 0,   aₙ = 0
```
  and the expansion contains only sine terms.
- Reduced coefficients:
  TEXT
```
     bₙ = (2/L)∫[0 to L] f(x)sin(nπx/L) dx
```

Neither parity: If (f(-x)) is neither (f(x)) nor (-f(x)), both cosine and sine coefficients generally need evaluation.
Decomposition: Every suitable function on a symmetric interval can be separated into even and odd parts:
TEXT
```
  fₑ(x) = {f(x) + f(-x)}/2
  fₒ(x) = {f(x) - f(-x)}/2
  f(x) = fₑ(x) + fₒ(x)
```
where (f_e) produces the cosine terms and (f_o) produces the sine terms.

VI. One-Sided Data — Sine and Cosine Extensions

A. Half range series

A half-range series represents a function given only on ((0,L)) by choosing either an even or odd extension to ((-L,L)).

Half-range cosine series: Extend (f) evenly by setting (F(-x)=F(x)).
- Resulting series:
  TEXT
```
     f(x) ~ a₀/2 + Σ[n=1 to ∞] aₙcos(nπx/L)
```
- Coefficients:
  TEXT
```
     a₀ = (2/L)∫[0 to L] f(x) dx

     aₙ = (2/L)∫[0 to L] f(x)cos(nπx/L) dx
```
- Boundary interpretation: Cosine modes naturally have zero derivative at (x=0) and (x=L), making them useful for insulated or zero-flux boundary conditions.
Half-range sine series: Extend (f) oddly by setting (F(-x)=-F(x)).
- Resulting series:
  TEXT
```
     f(x) ~ Σ[n=1 to ∞] bₙsin(nπx/L)
```
- Coefficients:
  TEXT
```
     bₙ = (2/L)∫[0 to L] f(x)sin(nπx/L) dx
```
- Boundary interpretation: Every sine mode vanishes at (x=0) and (x=L), making the basis appropriate for fixed-end, zero-temperature, or homogeneous Dirichlet boundary conditions.

Explicit contrast: The cosine series assumes mirror symmetry across (x=0), whereas the sine series assumes sign-changing symmetry. They represent the same original data on ((0,L)) but generally differ outside that interval.
Endpoint convergence: The selected extension determines convergence at (0), (L), and their periodic repetitions. An odd extension may introduce a jump at an endpoint if the given function does not approach zero there.
Application: Half-range expansions are central to separation-of-variables solutions for heat conduction, wave motion, and potential problems when data are supplied only over a physical interval rather than over a symmetric period.