---
title: "Fourier transform vs Fourier series"
subject: "Applied Mathematics"
chapter: 4
key: "semester_4/applied_mathematics:fourier-transform-vs-series"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-transform-vs-series
site: BCT Engineering
---

# Fourier transform vs Fourier series

Applied Mathematics · Chapter 4: Fourier Transform

- Asked in: 2073 Magh, 1.5+1.5+2 marks
- Video: [The Fourier Series and Fourier Transform Demystified](https://www.youtube.com/watch?v=mgXSevZmjPc) — Up and Atom
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-transform-vs-series
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-4

## Question

Define Fourier transform of a function. How does it differ from Fourier series? Support your answer with suitable example.

## Short answer

Fourier transform: F(s)=∫_{-∞}^{∞} f(x) e^{-isx} dx, inverse (1/2π)∫ F(s) e^{isx} ds. Series represents periodic functions with discrete spectrum; transform handles non-periodic functions giving continuous spectrum. Example: transform of e^{-|ax|} is 2a/(a²+s²); a periodic function uses series.

## Answer

#### Fourier transform

For a non-periodic, absolutely integrable function f(x), the Fourier transform is

**F(s) = ∫_{−∞}^{∞} f(x) e^{−isx} dx**

and the original function is recovered by the inverse transform

f(x) = (1/2π) ∫_{−∞}^{∞} F(s) e^{isx} ds.

(Some texts put 1/√(2π) in front of both formulas.) F(s) gives the amount of each frequency s in f(x), a continuous spectrum.

#### Fourier series vs Fourier transform

| Point | Fourier series | Fourier transform |
|---|---|---|
| Applies to | Periodic functions | Non-periodic functions (or the limit of infinite period) |
| Domain | Interval of one period [−L, L] | Whole line (−∞, ∞) |
| Spectrum | Discrete (frequencies nπ/L) | Continuous (all s) |
| Representation | Sum Σ cₙ e^{inπx/L} | Integral ∫ F(s) e^{isx} ds |
| Coefficients | Numbers aₙ, bₙ or cₙ | A function F(s) |
| Condition | Dirichlet conditions | f absolutely integrable |

The transform is the limit of the series when the period 2L → ∞: the discrete frequencies come closer and the sum turns into an integral.

#### Examples

**Fourier series.** f(x) = x on (−π, π), period 2π:

x = 2 [ sin x − (sin 2x)/2 + (sin 3x)/3 − … ]

**Fourier transform.** f(x) = e^{−|x|} (not periodic):

F(s) = ∫ e^{−|x|} e^{−isx} dx = 2/(1 + s²)

The series gives a list of coefficients for a periodic function; the transform gives a continuous function of s for a single non-repeating pulse.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. When is Fourier series used vs transform?
   A. Series for non-periodic, transform for periodic
   B. Both only for periodic
   C. Series for periodic functions, transform for non-periodic
   D. Both only for odd functions
   Answer: C. Transform handles single pulses on the whole line. (Key point: Applicability)

2. How do spectra differ?
   A. Series continuous; transform discrete
   B. Both discrete
   C. Both continuous
   D. Series gives discrete frequencies; transform gives continuous spectrum
   Answer: D. Frequencies n pi/L are discrete. (Key point: Spectrum)

3. How does transform arise from the series?
   A. As the period 2L tends to infinity
   B. As L tends to 0
   C. By making f odd
   D. By taking real parts
   Answer: A. The sum becomes an integral. (Key point: Relation)

4. What is the transform of e^{-|x|}?
   A. 1/(1 + s^2)
   B. 2/(1 + s^2)
   C. 2s/(1 + s^2)
   D. 2/(1 - s^2)
   Answer: B. 2 int from 0 e^{-x} cos sx dx. (Key point: Example)

## Related in Applied Mathematics

- [Define z-transform; differ from Fourier transform](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Az-transform-vs-fourier) — Applied Mathematics, chapter 2
- [Fourier transform of 1-x²; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-1-x2-evaluate) — Applied Mathematics, chapter 4, asked 2×
- [Complex Fourier series; verify convolution](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-series-complex-convolution) — Applied Mathematics, chapter 4

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