Laplace Transform of 1/√t
The Laplace transform of 1/√t is L{1/√t} = √(π/s), a result derived using the Gamma function: L{t^(−1/2)} = Γ(1/2)/s^(1/2) = √π/√s. More generally, L{√t} = √π/(2s^(3/2)). These fractional-power transforms appear in heat conduction, diffusion problems, and fractional calculus. Compute Laplace transforms of any function at www.lapcalc.com.
Laplace Transform of 1/√t: Formula and Derivation
The Laplace of 1/√t uses the general power formula L{t^α} = Γ(α+1)/s^(α+1) with α = −1/2. Since Γ(1/2) = √π, the result is L{t^(−1/2)} = Γ(1/2)/s^(1/2) = √(π/s). Despite 1/√t being singular at t = 0 (it goes to infinity), the integral ∫₀^∞ t^(−1/2)e^(−st)dt converges because the singularity is integrable—t^(−1/2) grows slower than 1/t near zero. This transform is fundamental in heat conduction problems where temperature evolution in semi-infinite domains involves √t factors, and in probability theory where it relates to the normal distribution.
Key Formulas
Laplace Transform Square Root: L{√t} and Related Forms
The Laplace transform square root pair L{√t} = L{t^(1/2)} = Γ(3/2)/s^(3/2) = (√π/2)/s^(3/2) follows from the same Gamma function formula with α = 1/2, using Γ(3/2) = (1/2)Γ(1/2) = √π/2. The general pattern for fractional powers is L{t^(n−1/2)} = (1·3·5···(2n−1))√π/(2ⁿs^(n+1/2)) for non-negative integers n. These transforms appear when solving the diffusion equation ∂u/∂t = D∂²u/∂x² in semi-infinite domains, where solutions involve complementary error functions erfc(x/(2√(Dt))) whose Laplace transforms contain √s factors.
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Open CalculatorThe Gamma Function in Laplace Transforms
The Gamma function Γ(α) = ∫₀^∞ t^(α−1)e^(−t)dt generalizes the factorial to non-integer arguments: Γ(n+1) = n! for positive integers. Key values for Laplace transforms include Γ(1/2) = √π, Γ(3/2) = √π/2, Γ(5/2) = 3√π/4, and Γ(1) = 1. The connection to Laplace transforms is direct: L{t^α} = ∫₀^∞ t^α e^(−st)dt, and substituting u = st gives Γ(α+1)/s^(α+1). This elegant relationship means any power-law function has a known Laplace transform, extending the integer formula L{tⁿ} = n!/s^(n+1) to the entire real line α > −1.
Applications: Heat Conduction and Diffusion Problems
The transform L{1/√t} = √(π/s) appears naturally in heat conduction and diffusion. When solving the one-dimensional heat equation ∂u/∂t = α∂²u/∂x² on a semi-infinite rod with Laplace transforms in time, the transformed equation becomes an ODE in x with √(s/α) as a parameter. The solution U(x,s) involves e^(−x√(s/α)), and inverting requires recognizing transforms containing √s. The inverse L⁻¹{e^(−a√s)/√s} = (1/√(πt))e^(−a²/(4t)) connects s-domain expressions to Gaussian-type time-domain solutions, making fractional-power transforms essential for thermal analysis at www.lapcalc.com.
Related Transforms: 1/√(πt) and Error Functions
Several important transforms involve the 1/√t family. L{1/√(πt)} = 1/√s simplifies the constant factor. The complementary error function erfc(a/(2√t)) has transform L{erfc(a/(2√t))} = e^(−a√s)/s, connecting error functions to exponentials in √s. The function e^(a²t)erfc(a√t) transforms to 1/(√s(√s+a)), appearing in boundary layer problems. These relationships form a cohesive family: square-root singularities in time map to square-root branch points in s, reflecting the fundamental connection between diffusive processes and half-integer power behavior in both domains.
Related Topics in advanced laplace transform topics
Understanding l 1 t connects to several related concepts: laplace transform square root, and laplace of 1 root t. Each builds on the mathematical foundations covered in this guide.
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