---
title: "Half range sine and cosine series: eˣ"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:half-range-sine-and-cosine-series"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-and-cosine-series
site: BCT Engineering
---

# Half range sine and cosine series: eˣ

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2079 Baishakh, 2+3 marks
- Video: [Half Range sine series f(x)=e^x](https://www.youtube.com/watch?v=BwjvnPvIdWg) — Btech Maths Hub
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-and-cosine-series
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the half range sine as well as cosine series for f(x) = eˣ in (0, 2).

## Short answer

Half-range sine series: e^x = 2π Σ_{n=1}^∞ [n(1−(−1)^n e^2)/(n^2π^2+4] sin(nπx/2). Half-range cosine series: e^x = (e^2−1)/2 + 4 Σ_{n=1}^∞ [((−1)^n e^2−1)/(n^2π^2+4] cos(nπx/2).

## Answer

**Half range sine series** on (0, L) is f(x) = Σ bₙ sin(nπx/L) with bₙ = (2/L)∫₀ᴸ f(x) sin(nπx/L) dx. Here L = 2, so bₙ = ∫₀² eˣ sin(nπx/2) dx.

Use ∫ eˣ sin kx dx = eˣ(sin kx − k cos kx)/(1 + k²) with k = nπ/2 (at x = 2: sin nπ = 0, cos nπ = (−1)ⁿ):

```text
bₙ = [ eˣ(sin kx − k cos kx)/(1 + k²) ]₀²
   = [ −k(−1)ⁿ e² + k ] / (1 + k²)
   = k[1 − (−1)ⁿ e²]/(1 + k²)
   = 2nπ[1 − (−1)ⁿ e²] / (n²π² + 4)
```

**Sine series**

eˣ = 2π Σ (n=1 to ∞) { n[1 − (−1)ⁿ e²] / (n²π² + 4) } sin(nπx/2),  0 < x < 2

**Half range cosine series** on (0, L) is f(x) = a₀/2 + Σ aₙ cos(nπx/L), with a₀ = (2/L)∫₀ᴸ f dx and aₙ = (2/L)∫₀ᴸ f cos(nπx/L) dx.

Use ∫ eˣ cos kx dx = eˣ(cos kx + k sin kx)/(1 + k²), k = nπ/2:

```text
a₀ = ∫₀² eˣ dx = e² − 1

aₙ = [ eˣ(cos kx + k sin kx)/(1 + k²) ]₀²
   = [ (−1)ⁿ e² − 1 ]/(1 + k²)
   = 4[(−1)ⁿ e² − 1] / (n²π² + 4)
```

**Cosine series**

eˣ = (e² − 1)/2 + 4 Σ (n=1 to ∞) { [(−1)ⁿ e² − 1] / (n²π² + 4) } cos(nπx/2),  0 < x < 2

Both series represent eˣ on (0, 2); the sine series converges slowly because its odd extension has jumps at the end points, while the cosine series (an even extension is continuous) converges faster. A numerical check at x = 1 gives 2.718 = e from both series.

## Check yourself

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

1. For e^x on (0, 2), what is L and how does bn simplify?
   A. L = 2, bn = integral from 0 to 2 of e^x sin(n pi x/2) dx
   B. L = 2 pi, bn = integral of e^x sin nx dx
   C. L = 1, bn = 2 times the integral
   D. L = 2, bn = integral of e^x cos(n pi x/2) dx
   Answer: A. 2/L = 1 for L = 2. (Key point: L value)

2. At x = 2, what are sin kx and cos kx with k = n pi/2?
   A. 0 and 1
   B. 0 and (-1)^n
   C. 1 and 0
   D. (-1)^n and 0
   Answer: B. kx = n pi. (Key point: Cos at x = 2)

3. What is a0 for the cosine series?
   A. e^2 + 1
   B. 2(e^2 - 1)
   C. (e^2 - 1)/2
   D. e^2 - 1
   Answer: D. a0 = (2/2) integral of e^x from 0 to 2. (Key point: Cosine series a0)

4. Why does the cosine series converge faster?
   A. It has no constant term
   B. Its even extension is continuous
   C. Its coefficients are zero
   D. The odd extension is continuous
   Answer: B. The sine series' odd extension has jumps at the ends. (Key point: Convergence)

## Close variants set in other exams

- [Half range sine series: e^(2x)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-2x) — Mathematics III, chapter 4, asked 6×

## Related in Mathematics III

- [Half range cosine series: πx − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-cosine-series-x-x2) — Mathematics III, chapter 4, asked 3×
- [Half range cosine series](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-cosine-series) — Mathematics III, chapter 4, asked 3×
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- [Half range cosine series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-cosine-series-e) — Mathematics III, chapter 4, asked 3×
- [Half range cosine series: x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-cosine-series-x) — Mathematics III, chapter 4, asked 3×
- [Half range cosine series: x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-cosine-series-x-two) — Mathematics III, chapter 4, asked 2×

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