Laplace Transform of Heaviside Step Function

Theorem

Let $\map {u_c} t$ denote the Heaviside step function:

$\map {u_c} t = \begin {cases} 1 & : t > c \\ 0 & : t < c \end {cases}$

The Laplace transform of $\map {u_c} t$ is given by:

$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} } s$

for $\map \Re s > c$.

Proof 1

$\ds \laptrans {\map {u_c} t}$

$=$

$\ds \int_0^{\to +\infty} \map {u_c} t e^{-s t} \rd t$

Definition of Laplace Transform

$\ds $

$=$

$\ds \int_0^c \map {u_c} t e^{-s t} \rd t + \int_c^{\to +\infty} \map {u_c} t e^{-s t} \rd t$

Sum of Integrals on Adjacent Intervals for Integrable Functions

$\ds $

$=$

$\ds \int_0^c 0 \times e^{-s t} \rd t + \int_c^{\to +\infty} 1 \times e^{-s t} \rd t$

Definition of Heaviside Step Function

$\ds $

$=$

$\ds \int_c^{\to +\infty} e^{-s t} \rd t$

$\ds $

$=$

$\ds \lim_{L \mathop \to +\infty} \int_c^L e^{-s t} \rd t$

Definition of Improper Integral

$\ds $

$=$

$\ds \lim_{L \mathop \to +\infty} \intlimits {\dfrac {e^{-s t} } {-s} } c L$

Primitive of $e^{a x}$

$\ds $

$=$

$\ds \lim_{L \mathop \to +\infty} \dfrac {e^{-s L} } {-s} - \dfrac {e^{-s c} } {-s}$

$\ds $

$=$

$\ds 0 + \dfrac {e^{-s c} } s$

simplification

Hence the result.

$\blacksquare$

Proof 2

$\ds \laptrans 1$

$=$

$\ds \dfrac 1 s$

Laplace Transform of 1

$\ds \leadsto \ \ $

$\ds \laptrans {1 \times \map {u_c} t}$

$=$

$\ds \dfrac 1 s \times e^{-c s}$

Second Translation Property of Laplace Transforms

$\ds $

$=$

$\ds \dfrac {e^{-s c} } s$

simplification

Hence the result.

$\blacksquare$

Sources

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $11$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $114.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.138$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of Special Laplace Transforms: $33.138.$