11. Fourier Transform in Complex Analysis

Fourier Transform plays a powerful role in both applied and theoretical mathematics. When combined with complex analysis, it becomes even more elegant and effective, especially when computing integrals and understanding the structure of functions in the frequency domain.

This section explains:

• Use of contour integration
• Fourier inversion formula using residues
• Interpretation of signals in the Fourier domain

1. Review of Fourier Transform

The Fourier Transform (FT) of a function f(t) is defined as:

F(w) = ∫ from -∞ to ∞ of [ f(t) * e^(-i * w * t) dt ]
    

Where:

• f(t): Function in the time domain
• F(w): Function in the frequency domain (Fourier domain)
• w: Angular frequency

The Inverse Fourier Transform is:

f(t) = (1 / 2π) * ∫ from -∞ to ∞ of [ F(w) * e^(i * w * t) dw ]
    

2. Why Use Complex Analysis?

Complex analysis helps in:

• Evaluating difficult integrals using complex variables
• Extending real integrals to the complex plane
• Using the Residue Theorem for evaluating integrals via poles

3. Contour Integration – Basics

In complex analysis, integrals are often evaluated along paths (contours) in the complex plane.

Contour: A directed curve in the complex plane.

Contour Integral:

∮ f(z) dz (around a closed path)
    

Residue Theorem

If f(z) is analytic inside a contour except for isolated singularities (poles), then:

∮ f(z) dz = 2πi × sum of residues at those poles inside the contour
    

4. Fourier Inversion Using Residues

To retrieve f(t) from F(w), we evaluate:

f(t) = (1 / 2π) * ∫ from -∞ to ∞ of [ F(w) * e^(i * w * t) dw ]
    

This integral can be evaluated using complex contour integration. The direction of closing the contour depends on the sign of t:

• If t > 0: Close the contour in the upper half-plane
• If t < 0: Close in the lower half-plane

Using Jordan’s Lemma, we argue the arc vanishes, leaving only the residues at the poles of the function F(w) * e^(i * w * t).

Example

Let F(w) = 1 / (w² + 1)

Pole locations: w = i, -i

If t > 0, close in the upper half-plane and take the residue at w = i:

Residue at w = i: e^(-t) / (2i)

Then:
f(t) = (1 / 2π) × 2πi × (e^(-t) / 2i) = e^(-t) / 2
    

This is the inverse Fourier Transform result using residue calculus.

5. Interpreting Signals in the Fourier Domain

The Fourier domain shows how much of each frequency exists in a signal.

• Sine wave → Single spike at that frequency
• Rectangular pulse → Spread over many frequencies (sinc function)
• Delta function → Constant in frequency (contains all frequencies)

Visualization:

• Time domain: How signal changes with time
• Frequency domain: Shows contribution of each frequency

Example from music:
- Time domain: The waveform of the sound
- Frequency domain: How much bass, mid, and treble is present

6. Deeper Understanding for Discovery

By mastering:

• Contour techniques
• Pole and singularity analysis
• Residue computation
• Fourier domain behavior

You’ll be in a better position to innovate and discover even better transforms like:

• Wavelet Transforms – Better for localized time-frequency analysis
• Laplace Transforms – Better for causal systems
• STFT (Short-Time Fourier Transform) – Good for non-stationary signals

This depth of understanding can help students or engineers design custom transforms, or even propose a new one better than the original Fourier Transform in specific cases.

In the next topic, we will explore windowed Fourier transforms (STFT), its applications, and how it improves upon classical Fourier analysis.

Fourier Transform: 20 Solved Examples (From Basic to Competent Mastery)

Each example below includes the question, step-by-step solution, explanation of every symbol, and reasoning for the formula used.

Example 1: Fourier Transform of f(t) = 1

Solution:
F(w) = ∫ from -∞ to ∞ of 1 × e^−i·w·t dt
This integral diverges normally, but in distribution theory:
F(w) = 2π·δ(w)
δ(w) is the Dirac delta function — infinite spike at w=0.

Example 2: Fourier Transform of f(t) = e^−a·|t|, a > 0

Split into two parts:
For t < 0: f(t) = e^a·t ; For t ≥ 0: f(t) = e^−a·t
Combine both integrals:
F(w) = 1 / (a + i·w) + 1 / (a − i·w) = 2a / (a² + w²)

Example 3: Rectangular Pulse (−T/2 ≤ t ≤ T/2)

F(w) = ∫ from −T/2 to T/2 of e^−i·w·t dt
= (2·sin(w·T/2)) / w = T·sinc(w·T/2)

Example 4: f(t) = cos(w₀·t)

F(w) = π [δ(w − w₀) + δ(w + w₀)]

Example 5: f(t) = sin(w₀·t)

F(w) = i·π [δ(w + w₀) − δ(w − w₀)]

Example 6: Gaussian f(t) = e^−a·t²

F(w) = √(π/a) · e^{−w² / (4a)}

Example 7: Scaling Property

If f(t) = e^−|t|, then f(bt) ⇨ F(w) = (1/|b|)·F(w/b)

Example 8: Time Shift

If f(t) ⇨ F(w), then f(t − t₀) ⇨ e^{−i·w·t₀} · F(w)

Example 9: Frequency Shift

e^i·w₀·t · f(t) ⇨ F(w − w₀)

Example 10: Differentiation

FT of df/dt = i·w·F(w)

Example 11: Convolution

FT of (f * g)(t) = F(w) · G(w)

Example 12: Inverse FT via Residue

Given F(w) = 1/(w² + 1), poles at w = ±i
Residue at w = i = e^−t / (2i)
Then f(t) = e^−t/2

Example 13: Pulse Train

Fourier Transform of Dirac comb in time is a Dirac comb in frequency

Example 14: Triangular Pulse

FT of triangular function is sinc²(w)

Example 15: Modulated Rectangular Pulse

f(t) = cos(w₀·t) · rect(t/T) ⇨
F(w) = (T/2) [sinc((w − w₀)·T/2) + sinc((w + w₀)·T/2)]

Example 16: Unit Step Function u(t)

Fourier Transform doesn’t exist regularly, Laplace is used instead.

Example 17: Delta Shift

f(t) = δ(t − t₀) ⇨ F(w) = e^{−i·w·t₀}

Example 18: LTI System

If h(t) is impulse response, output Y(w) = H(w)·X(w)

Example 19: Complex Exponential

f(t) = e^i·w₀·t ⇨ F(w) = 2π·δ(w − w₀)

Example 20: Complex Inverse FT via Residue

Given F(w) = w / (w² + 1)
Poles: w = ±i, residue at w = i is i / (2i) = 1/2
f(t) = e^−t / 2 for t > 0

Conclusion:
These examples were selected to gradually build your understanding of Fourier Transform, properties, inversion, and how complex analysis (residue theorem) enhances our ability to solve otherwise difficult integrals. Each formula is explained and justified. Mastering these allows one to deeply explore and even derive better transforms in specialized domains.