What is a Laplace transform?

A Laplace transform is an integral transform that converts a time-domain function $f(t)$ into a complex frequency-domain function $F(s)$, simplifying differential equations into algebraic ones.

What is the formula for a Laplace transform?

The Laplace transform of $f(t)$ is $\mathcal{L}\{f(t)\} = \int_0^\infty f(t) e^{-st} \, dt$.

What is the Laplace transform of a constant C?

The Laplace transform of a constant $C$ is $C/s$, which is valid for $s > 0$.

What is the Laplace transform of t^n?

The Laplace transform of $t^n$ (for positive integers $n$) is $n! / s^{n+1}$.

What is the Laplace transform of e^(at)?

The Laplace transform of $e^{at}$ is $1 / (s - a)$, which is valid for $s > a$.

What is the Laplace transform of sin(wt)?

The Laplace transform of $\sin(wt)$ is $w / (s^2 + w^2)$.

What is the Laplace transform of cos(wt)?

The Laplace transform of $\cos(wt)$ is $s / (s^2 + w^2)$.

What is an inverse Laplace transform?

The inverse Laplace transform converts an s-domain function $F(s)$ back into a time-domain function $f(t)$, typically using partial fraction expansion and look-up tables.

Why are poles important in Laplace transforms?

Poles are the roots of the denominator of $F(s)$. Their values determine the exponential behavior of the system's response (e.g., decay, oscillation, or growth).

What does system stability mean in the s-plane?

A system is stable if all its poles lie in the left half of the s-plane (negative real parts), which ensures the time-domain response decays to zero.

How does differentiation in time affect the Laplace transform?

Differentiation in time corresponds to multiplication by $s$ in the s-domain, minus the initial conditions: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$.

What is the transfer function?

The transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero.

math calculator

Laplace & Inverse Laplace Transform Solver

Compute Laplace transforms for t-domain functions and inverse Laplace transforms with steps.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Transform Direction

Numerator: As + B

Coefficient A

Constant B

Denominator: s² + bs + c

Coefficient b

Constant c

Results

Poles Category

Distinct Real Poles

Time Domain f(t)f(t) = 3.667e^{2.00t} -0.667e^{-1.00t}

Time Response Waveform

Time Response Waveform y(t) for t ≥ 0

Step-by-Step Solution

Given the s-domain function: \[F(s) = \frac{3s + 5}{s^2 -1s -2}\]

The denominator has two distinct real roots (poles): $s_1 = 2.000$ and $s_2 = -1.000$.

Decompose using partial fractions:

\[\frac{3s + 5}{(s - 2.000)(s - -1.000)} = \frac{K_1}{s - 2.000} + \frac{K_2}{s - -1.000}\]

Solve for residuals:

\[K_1 = \frac{3(2.000) + 5}{2.000 - -1.000} = 3.6667\]

\[K_2 = \frac{3(-1.000) + 5}{-1.000 - 2.000} = -0.6667\]

Apply inverse Laplace transform:

\[f(t) = 3.6667 e^{2.00t} -0.6667 e^{-1.00t}\]

Formula

Laplace Transform formulas

\mathcal{L}^{-1}\{F(s)\} = \text{Partial Fraction match}

Differentiation in the time-domain corresponds to scaling by s in the complex s-domain, converting differential equations to algebraic equations.

Step by step

How the calculation works

1Choose the conversion direction (Forward Laplace vs. Inverse Laplace).
2For Forward transforms, choose the base function and input its scaling parameters.
3For Inverse transforms, specify the polynomial coefficients of the rational s-domain expression.

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Decomposition

Partial Fractions

Solve polynomial factor decompositions.

Fourier Series

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math calculator

Laplace & Inverse Laplace Transform Solver

Compute Laplace transforms for t-domain functions and inverse Laplace transforms with steps.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Transform Direction

Numerator: As + B

Coefficient A

Constant B

Denominator: s² + bs + c

Coefficient b

Constant c

Results

Poles Category

Distinct Real Poles

Time Domain f(t)f(t) = 3.667e^{2.00t} -0.667e^{-1.00t}

Time Response Waveform

Time Response Waveform y(t) for t ≥ 0

Step-by-Step Solution

Given the s-domain function: \[F(s) = \frac{3s + 5}{s^2 -1s -2}\]

The denominator has two distinct real roots (poles): $s_1 = 2.000$ and $s_2 = -1.000$.

Decompose using partial fractions:

\[\frac{3s + 5}{(s - 2.000)(s - -1.000)} = \frac{K_1}{s - 2.000} + \frac{K_2}{s - -1.000}\]

Solve for residuals:

\[K_1 = \frac{3(2.000) + 5}{2.000 - -1.000} = 3.6667\]

\[K_2 = \frac{3(-1.000) + 5}{-1.000 - 2.000} = -0.6667\]

Apply inverse Laplace transform:

\[f(t) = 3.6667 e^{2.00t} -0.6667 e^{-1.00t}\]

Formula

Laplace Transform formulas

\mathcal{L}^{-1}\{F(s)\} = \text{Partial Fraction match}

Differentiation in the time-domain corresponds to scaling by s in the complex s-domain, converting differential equations to algebraic equations.

Step by step

How the calculation works

1Choose the conversion direction (Forward Laplace vs. Inverse Laplace).
2For Forward transforms, choose the base function and input its scaling parameters.
3For Inverse transforms, specify the polynomial coefficients of the rational s-domain expression.

Related guides

Learn more about this calculation

Guide

Analyze steady-state frequency harmonics.