---
title: "Laplace transforms of functions 1/√t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-1-t"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-1-t
site: BCT Engineering
---

# Laplace transforms of functions 1/√t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2080 Baishakh, 1+2+2 marks; 2078 Bhadra, 1+2+2 marks (asked 2 times)
- 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-1-t
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

State the existence condition for the Laplace transform. Obtain the Laplace transform of a) 1/√t  b) (1 − cos 2t)/t.

## Short answer

A sufficient condition: f piecewise continuous and of exponential order |f(t)| ≤ M e^(kt). L{1/√t} = √(π/s); L{(1-cos2t)/t} = (1/2) ln((s^2+4)/s^2), using integration of the transform.

## Answer

**Existence condition.** The Laplace transform L{f(t)} = ∫₀^∞ e^(−st) f(t) dt exists for s > k if f(t) is piecewise continuous on every finite interval [0, T] and is of exponential order, that is |f(t)| ≤ M e^(kt) for all t ≥ T. (These conditions are sufficient, not necessary.)

#### a) L{1/√t}
Use L{tⁿ} = Γ(n + 1)/s^(n+1) with n = −1/2:

```text
L{t^(−1/2)} = Γ(1/2)/s^(1/2) = √π / √s = √(π/s)
```

#### b) L{(1 − cos 2t)/t}
Use L{f(t)/t} = ∫ₛ^∞ F(u) du, with F(s) = L{1 − cos 2t} = 1/s − s/(s² + 4). (The limit of f(t)/t as t → 0 exists and equals 0.)

```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))
 = ½ ln((s² + 4)/s²)
```

**Answers:** a) √(π/s); b) 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. Laplace transform exists if f is:
   A. bounded and periodic
   B. differentiable everywhere
   C. piecewise continuous and of exponential order
   D. continuous with f(0)=0
   Answer: C. Sufficient conditions. (Key point: Existence)

2. L{1/sqrt t}=
   A. sqrt(s/pi)
   B. pi/s
   C. 1/sqrt s
   D. sqrt(pi/s)
   Answer: D. Gamma(1/2)=sqrt pi. (Key point: a)

3. L{f/t}=
   A. integral of F(u) from s to infinity
   B. derivative of F
   C. integral from 0 to s
   D. -F'(s)
   Answer: A. Division by t. (Key point: b method)

4. L{(1-cos2t)/t}=
   A. ln(s^2+4)
   B. (1/2) ln((s^2+4)/s^2)
   C. (1/2) ln(s^2/(s^2+4))
   D. arctan(2/s)
   Answer: B. Integral computed. (Key point: b)

## Close variants set in other exams

- [Laplace transforms of functions cos³2t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-cos32t) — Mathematics III, chapter 3, asked 4×
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