What is the Laplace transform of sinh(at)?

L{sinh(at)} = a/(s²−a²) for Re(s) > |a|. This is derived using sinh(at) = (e^(at)−e^(−at))/2 and the linearity of the Laplace transform. Note the minus sign in s²−a² (unlike s²+a² for sin), reflecting that sinh grows exponentially rather than oscillating.

How does L{tⁿ} extend to non-integer powers?

For non-integer α > −1, L{t^α} = Γ(α+1)/s^(α+1), where Γ is the Gamma function. For integers, Γ(n+1) = n!, recovering the standard formula. Notable examples include L{√t} = √π/(2s^(3/2)) and L{1/√t} = √(π/s).

Why do repeated poles produce polynomial terms in the time domain?

A repeated pole of order n at s = −a contributes a term proportional to t^(n−1)e^(−at)/(n−1)! in the time domain, since L⁻¹{1/(s+a)ⁿ} = t^(n−1)e^(−at)/(n−1)!. The polynomial factor t^(n−1) creates a transient that initially grows before the exponential decay dominates.

Laplace Transform of T 2

Quick Answer

The Laplace transform of t² is L{t²} = 2/s³, derived from the general formula L{tⁿ} = n!/s^(n+1) with n = 2. Similarly, the Laplace transform of sinh(at) is L{sinh(at)} = a/(s²−a²). These are fundamental transform pairs used in polynomial and hyperbolic function analysis. Compute any Laplace transform instantly at www.lapcalc.com.

Calculate this instantly on LAPLACE Calculator →

Laplace of t²: Derivation from First Principles

The Laplace of t² is computed directly from the definition: L{t²} = ∫₀^∞ t²e^(−st)dt. Applying integration by parts twice—first with u = t², dv = e^(−st)dt, then with u = 2t, dv = e^(−st)dt—yields L{t²} = 2/s³ for Re(s) > 0. This result also follows from the general power formula L{tⁿ} = n!/s^(n+1): with n = 2, we get 2!/s³ = 2/s³. The pattern continues: L{t³} = 6/s⁴, L{t⁴} = 24/s⁵. Each power of t adds one more factor of 1/s, reflecting that higher-degree polynomials grow faster and require stronger s-domain decay to converge.

Key Formulas

Laplace Transform of t² Combined with Other Functions

The Laplace transform of t² becomes more interesting when combined with exponentials and trigonometric functions. Using frequency shifting: L{t²e^(−at)} = 2/(s+a)³. Using the multiplication-by-t property L{t·f(t)} = −F′(s) applied twice: L{t²·f(t)} = F″(s). For example, L{t²sin(ωt)} = d²/ds²[ω/(s²+ω²)], which requires careful differentiation. These composite transforms arise in systems with polynomial-modulated signals and in the partial fraction expansion of transfer functions with repeated poles, where terms like A/(s+a)³ invert to (A/2)t²e^(−at).

Compute laplace transform of t 2 Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

Laplace of sinh(at): Hyperbolic Sine Transform

The Laplace of sinh is L{sinh(at)} = a/(s²−a²) for Re(s) > |a|. This follows from the definition sinh(at) = (e^(at) − e^(−at))/2 and linearity: L{sinh(at)} = (1/2)[1/(s−a) − 1/(s+a)] = (1/2)·2a/(s²−a²) = a/(s²−a²). Note the critical difference from the sine transform: L{sin(at)} = a/(s²+a²) has s²+a² in the denominator (poles on imaginary axis), while L{sinh(at)} = a/(s²−a²) has s²−a² (poles on real axis at s = ±a). This reflects that sinh grows exponentially while sin oscillates.

General Power Formula and the Gamma Function Extension

The formula L{tⁿ} = n!/s^(n+1) applies for non-negative integers n, but extends to non-integer powers using the Gamma function: L{t^α} = Γ(α+1)/s^(α+1) for α > −1. The Gamma function satisfies Γ(n+1) = n! for integers, providing seamless generalization. For example, L{t^(1/2)} = Γ(3/2)/s^(3/2) = (√π/2)/s^(3/2), and L{t^(−1/2)} = Γ(1/2)/s^(1/2) = √(π/s). These fractional-power transforms appear in diffusion problems, fractional calculus, and certain heat transfer solutions involving semi-infinite domains.

Polynomial Laplace Transforms in System Response Analysis

Polynomial Laplace transforms appear directly in system response analysis. When a transfer function has repeated poles, the partial fraction expansion includes terms like A/(s+a)², B/(s+a)³, etc., which invert to Ate^(−at), (B/2)t²e^(−at), and so on. The polynomial factor tⁿ indicates the multiplicity of the pole and produces characteristic responses that grow before decaying. In step response analysis at www.lapcalc.com, a double pole at s = −a produces terms proportional to te^(−at), while a triple pole adds t²e^(−at) terms. Understanding L{tⁿ} = n!/s^(n+1) is therefore essential for interpreting system behavior from transfer function pole structure.

Frequently Asked Questions

The Laplace transform of t² is 2/s³, following from the general formula L{tⁿ} = n!/s^(n+1) with n = 2. It can also be derived by integrating ∫₀^∞ t²e^(−st)dt twice by parts. This pair is essential for handling repeated poles in partial fraction expansions.

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →