Laplace Transform Standard Formulae

LAPLACE TRANSFORM
STANDARD FORMS

SJMATHTUBE | Exam-ready formula sheet for engineering mathematics

Pronunciation Laplace transforms

1 Definition and Notation Start here

The Laplace transform converts a function of \(t\) into a function of \(s\).

Input function \(f(t)\) Generally lowercase \(f, g, \dots\) are used.

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Transformed function \(F(s)\) Generally uppercase \(F, G, \dots\) are used.

\[ \mathcal{L}\{f(t)\}=F(s)=\int_{0}^{\infty} e^{-st}f(t)\,dt \]

The original function is written in terms of \(t\). Its Laplace transform is written in terms of \(s\).

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PART 2 Basics

PART 3 Standard Forms

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2 Basic Standard Forms Core

\[ \mathcal{L}\{1\}=\frac{1}{s} \]

\[ \mathcal{L}\{e^{at}\}=\frac{1}{s-a} \]

\[ \mathcal{L}\{e^{-at}\}=\frac{1}{s+a} \]

For \(e^{at}\), the denominator is \(s-a\). For \(e^{-at}\), the denominator is \(s+a\).

3 Trigonometric Forms Plus sign

\[ \mathcal{L}\{\sin at\}=\frac{a}{s^{2}+a^{2}} \]

\[ \mathcal{L}\{\cos at\}=\frac{s}{s^{2}+a^{2}} \]

For \(\sin at\) and \(\cos at\), the denominator has plus: \(s^{2}+a^{2}\).

4 Hyperbolic Forms Minus sign

\[ \mathcal{L}\{\sinh at\}=\frac{a}{s^{2}-a^{2}} \]

\[ \mathcal{L}\{\cosh at\}=\frac{s}{s^{2}-a^{2}} \]

For \(\sinh at\) and \(\cosh at\), the denominator has minus: \(s^{2}-a^{2}\).

5 Power of \(t\) Very important

\[ \mathcal{L}\{t^{n}\}=\frac{n!}{s^{n+1}},\quad n\in\mathbb{N} \]

\[ \mathcal{L}\{t^{n}\}=\frac{\Gamma(n+1)}{s^{n+1}},\quad n>-1 \]

For most engineering mathematics questions, \(\mathcal{L}\{t^{n}\}=\frac{n!}{s^{n+1}}\) is the formula used most often.

6 Memory Aid Remember fast

Sine and Sinh \(\sin at\) and \(\sinh at\) do not take \(s\) on top. Both take \(a\) on top.

Cos and Cosh \(\cos at\) and \(\cosh at\) both take \(s\) on top.

Trig forms \(\sin at\) and \(\cos at\) use \(s^{2}+a^{2}\).

Hyperbolic forms \(\sinh at\) and \(\cosh at\) use \(s^{2}-a^{2}\).

7 Common Mistake Careful

Do not mix \(\sin at\) with \(\sinh at\).

Trigonometric \[ \mathcal{L}\{\sin at\}=\frac{a}{s^{2}+a^{2}} \]

Hyperbolic \[ \mathcal{L}\{\sinh at\}=\frac{a}{s^{2}-a^{2}} \]

LAPLACE TRANSFORM STANDARD FORMS

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LAPLACE TRANSFORM
STANDARD FORMS