Laplace Transform Solver (f(t) to F(s))

Laplace Transform Solver

Convert time-domain functions $f(t)$ to s-domain $F(s)$ instantly.
Supports standard functions, exponentials, and trigonometric shifts.

Input Function f(t)

Enter function of ‘t’ (e.g., sin(2t), e^(-3t))

Try Examples: t^2 sin(3t) e^(-2t) t*e^(-t) Damped Sin Unit Step

Definition

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

The Laplace transform converts differential equations in the time domain into algebraic equations in the s-domain (complex frequency domain).

Result F(s)

Standard Table

$$ F(s) = \frac{s+2}{(s+2)^2 + 9} $$

Property Applied: Frequency Shift Theorem (First Shift)

Time Domain f(t)

Causal Signal (t ≥ 0)

Understanding the Laplace Transform

The Laplace Transform is a powerful integral transform used to switch a function from the time domain ($t$) to the s-domain ($s$). It is especially useful for solving linear ordinary differential equations (ODEs) because it turns derivatives into multiplication by $s$, transforming calculus problems into algebra problems.

Key Properties

Linearity: $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$.
First Shift Theorem (Frequency Shift): Multiplying by an exponential in time shifts the s-domain variable. $$ \mathcal{L}\{e^{at}f(t)\} = F(s-a) $$ This is why inputting e^(-2t)cos(3t) shifts the cosine transform from $s/(s^2+9)$ to $(s+2)/((s+2)^2+9)$.
Differentiation in s: Multiplying by $t$ in time corresponds to differentiating in s. $$ \mathcal{L}\{t f(t)\} = -\frac{d}{ds}F(s) $$

Region of Convergence (ROC)

The integral definition only converges for certain values of $s$. For most causal signals (signals that start at $t=0$), the ROC is the region in the complex plane to the right of the rightmost pole (Re($s$) > $\sigma$).

FAQ

Why use Laplace instead of Fourier?

While Fourier analysis restricts signals to be stable (integrable), Laplace introduces the convergence factor $e^{-\sigma t}$, allowing us to analyze unstable systems and transient responses that grow over time.

What is ‘s’?

The variable ‘s’ is a complex number, usually written as $s = \sigma + j\omega$. The real part $\sigma$ relates to exponential decay/growth, and the imaginary part $\omega$ relates to sinusoidal oscillation frequency.

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