# Why are the limits of integration set as they are for the Laplace Transform?

Mathematics Asked by jonathan x on January 5, 2022

Is there a reason for the Laplace Transform starting at zero? Could the transform go from -1 to ∞ or 1 to ∞? I understand that the upper bound is for the sake of convergence; however, the lower bound seems fairly arbitrary. What does the Transform gain from starting at zero?

## One Answer

For a physicist's or engineer's perspective, let's examine what is probably the most common application of Laplace transforms in the real world. Suppose that $$x(t)$$, the position of a particle at time $$t$$ is given by the differential equation, $$m frac{d^2 x}{dt^2} + gamma frac{dx}{dt} + k x = F(t),$$ where $$F(t)$$ is the force applied on the particle at time $$t$$. ($$m$$, $$gamma$$ and $$k$$ represent the particle's mass, the friction and the strength of the restorative force respectively.)

Furthermore, we suppose that when $$t < 0$$, the force is zero and the system is stationary. The force is only "turned on" at time $$t = 0$$. Therefore, $$x(t)$$ and $$F(t)$$ are non-zero only for $$t geq 0$$, we only really need to solve the differential equation for $$t geq 0$$. (The initial conditions at $$t = 0$$ are $$x(0) = 0$$ and $$frac{dx}{dt}(0) = 0$$.)

We can solve this equation using Laplace transforms. Taking Laplace transforms of both sides, the differential equation turns into an algebraic equation which is easier to solve. Once we have solved this algebraic equation, we invert the Laplace transform, giving the final answer, $$x(t) = mathcal L^{-1} left[ frac{mathcal L[F](s)}{ms^2 + gamma s + k}right].$$

But the thing I want to draw your attention to is the fact that, in the context of this physics problem, the $$x(t)$$ and the $$F(t)$$ are only non-zero when $$t geq 0$$. So when you define their Laplace transforms, it is only natural to integrate over $$t geq 0$$: $$mathcal L[x](s) := int_0^infty e^{-ts} x(t) dt, mathcal L[F](s) := int_0^infty e^{-ts} F(t) dt.$$ Integrating over the whole of $$mathbb R$$ is pointless, seeing that $$x(t)$$ and $$F(t)$$ are zero when $$t < 0$$ anyway! So in the context of this physics problem, it is natural to define the Laplace transforms using integrals over $$[0, infty)$$.

Answered by Kenny Wong on January 5, 2022

## Related Questions

### How can I show the quotient of the $k$th partial sums of $sumlimits_{n=1}^{k} n$ and $sumlimits_{n=1}^{k} n^2$ is $frac{3}{2k+1}$?

2  Asked on March 8, 2021 by danoram

### Equation related to the curve $x^4+3kx^3+6x^2+5$

2  Asked on March 8, 2021

### $f$ is integrable & $int ^a _b f= beta iff forall epsilon >0 exists mathbb{P}$ partition such as $U(f,P)-epsilon < beta < L(f,P)+ epsilon$

1  Asked on March 7, 2021 by juju9708

### Let $S = {1,2,3,4}. X, Y in mathcal{P}(S)$ and R be a relation $R(X,Y): |X cap Y| = 1$. Is the relation R transitive?

1  Asked on March 6, 2021 by edohedo

### A presheaf can be seen as a contravariant functor

0  Asked on March 6, 2021 by rising_sea

### Prove that $int_0^b x^3 dx = frac{b^4}{4}$

0  Asked on March 6, 2021 by dansidorkin

### Doubts on the proof regarding uniform convergence on complex plane

0  Asked on March 5, 2021 by able20

### Connection between vector space isomorphisms and dimensions

1  Asked on March 5, 2021 by pedro-mariz

### Determine if $n$ could be represented by a quadratic form of discriminant $d$

1  Asked on March 4, 2021 by jibber032394

### Convergence of Bernoulli distribution using Central Limit Theorem

1  Asked on March 4, 2021 by big_golfuniformindia

### Example where an inverse function does not equal the elements

2  Asked on March 4, 2021 by mc5555

### $int frac{(e^z)}{z-pi i} dz$, if C is the ellipse |z – 2| + |z+2| = 6

2  Asked on March 4, 2021 by dip

### Finding the volume of a region using spherical coordinates

0  Asked on March 3, 2021 by rmdnusr

### Derive Greens function 1D heat equation with space dependent material parameters

0  Asked on March 2, 2021

### Sketch the solid described by the given inequalities.

1  Asked on March 2, 2021 by 2316354654

### Nonlinear ODE with discontinuous dynamic

0  Asked on March 1, 2021 by briantag

### Finding matrices whose multiplication does not change the trace

2  Asked on March 1, 2021 by hitesh

### Find the Laurent Series expansion for $displaystyle{f(z) = ze^{1/(z-1)}}$ valid for $left|z-1right|> 0$

1  Asked on March 1, 2021 by ariana-tibor

### Central Limit Theorem and Lightbulbs

1  Asked on March 1, 2021 by maksymilian5275

### A little help with elementary set intuition, specifically on intersection

2  Asked on March 1, 2021 by stephan-psaras

### Ask a Question

Get help from others!

© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir