---
title: "Laplace transforms of functions"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-two
site: BCT Engineering
---

# Laplace transforms of functions

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2073 Chaitra, 1+1+3 marks
- Video: [Intro to the Laplace Transform & Three Examples](https://www.youtube.com/watch?v=KqokoYr_h1A) — Dr. Trefor Bazett
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Define the Laplace transform of a function and state the criteria of existence of the Laplace transform of a function. Find the Laplace transform of f(t) = (1 − cos 2t)/t.

## Short answer

The Laplace transform is L[f(t)] = ∫_0^∞ e^(-st) f(t) dt. Existence requires f piecewise continuous on [0, A] and exponential order |f(t)| ≤ M e^(kt). For (1-cos 2t)/t, apply L[f/t] = ∫_s^∞ F(u) du to get ½ ln((s^2+4)/s^2).

## Answer

#### Definition
Let f(t) be defined for t ≥ 0. The Laplace transform of f(t) is

```text
L[f(t)] = F(s) = ∫₀^∞ e^(−st) f(t) dt
```

provided the integral converges for some values of s.

#### Conditions of existence
L[f(t)] exists for s > k if
1. f(t) is piecewise continuous on every finite interval [0, A], and
2. f(t) is of exponential order: |f(t)| ≤ M e^(kt) for all t ≥ T, for some constants M, k, T.

(These are sufficient conditions.)

#### L[(1 − cos 2t)/t]
Use L[f(t)/t] = ∫ₛ^∞ F(u) du. Here F(u) = 1/u − u/(u² + 4).

```text
∫ₛ^∞ [1/u − u/(u² + 4)] du = [ ln u − ½ ln(u² + 4) ]ₛ^∞
                            = [ ln( u/√(u² + 4) ) ]ₛ^∞
                            = 0 − ln( s/√(s² + 4) )
```

**Answer: ½ ln((s² + 4)/s²) = ln(√(s² + 4)/s)**

## Check yourself

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

1. L[f(t)] is defined as
   A. integral from -infinity to infinity of e^(-st) f(t) dt
   B. integral from 0 to infinity of e^(st) f(t) dt
   C. integral from 0 to 1 of e^(-st) f(t) dt
   D. integral from 0 to infinity of e^(-st) f(t) dt
   Answer: D. One-sided with decaying exponential. (Key point: Definition)

2. Sufficient conditions are
   A. piecewise continuity and exponential order
   B. f must be periodic
   C. f must be bounded and differentiable
   D. f(0) = 0
   Answer: A. |f| <= M e^(kt). (Key point: Existence conditions)

3. L[(1 - cos 2t)/t] uses
   A. -F'(s)
   B. integral of F(u) from s to infinity
   C. F(s)/s
   D. F(s-2)
   Answer: B. Division by t. (Key point: Method)

4. L[(1 - cos 2t)/t] =
   A. (1/2) ln(s^2/(s^2+4))
   B. ln((s^2+4)/s^2)
   C. (1/2) ln((s^2+4)/s^2)
   D. (1/2) ln((s+2)/s)
   Answer: C. -ln(s/sqrt(s^2+4)). (Key point: Result)

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