Laplace Transform

Laplace transform is an integral transform used in mathematics and engineering to convert a function of time f(t) into a function of a complex variable s, denoted as F(s), where s = \sigma+\iota\omega.

Let us assume f(t) is a function, be it a real or complex function of the variable t>0, where t is time. Then, the Laplace transform F(s) of f(t) is the complex function defined for s\in \Complex, given by:

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

where s = \sigma + i\omega.

Standard Notation:

If a function of t is indicated as f(t), f(t),g(t), or y(t), then their respective Laplace transforms are represented by F(s), G(s) and Y(s). Besides the notation F(s), we can also use \mathcal{L}\{f(t)\} or \mathcal{L}\{f\}(s).

Laplace transforms of some elementary functions:

Existence of the Laplace Transform

Here are some definitions before delving into the sufficient conditions for the existence of the Laplace transform:

• Sectional Continuity: A function is said to be sectionally or piecewise continuous in an interval t_1\le t\le t_2 if that interval can be subdivided into a finite number of subintervals, in each of which the function is continuous and has finite left- and right-hand limits.
• Functions of Exponential Order: If real constants k>0 and \gamma exist such that for all t>N, the function f(t) satisfies the condition |f(t)|\le ke^{\gamma t}, then f(t) is said to be of exponential order \bm\gamma.

Sufficient Condition for Existence of Laplace Transform:

If f(t) is sectionally continuous in every finite interval 0\le t\le N and of exponential order \bm\gamma for \bm{t>N}, then its Laplace transform F(s) exists for all \bm{s>\gamma}.

Important Properties of Laplace Transformation:

• Linearity
• Shifting
• Change of Scale
• Laplace Transforms of Derivatives
• Laplace Transforms of Integrals
• Multiplication by t^n
• Division by t
• Laplace Transform of a Periodic function
• Behavior of F(s) as s\to\infty
• Initial value theorem
• Final value theorem
• Convolution theorem for Laplace transform

Linearity

If c_1 and c_2 are two constants, while f_1(t) and f_2(t) are functions, and F_1(s) and F_2(s) are their respective Laplace transforms, then:

\begin{aligned}\mathcal{L}\{c_1f_1(t) +c_2f_2(t)\} &= c_1\mathcal{L}\{f_1(t)\}+c_2\mathcal{L}\{f_2(t)\}\\&=c_1F_1(s) +c_2F_2(s)\end{aligned}

Shifting

It constitutes of two properties

1. First shifting property: If \mathcal{L}\{f(t)\}=F(s), then \mathcal{L}\{e^{at}f(t)\}=F(s-a).
2. Second Shifting property: If \mathcal{L}\{f(t)\}=F(s) and g(t)=\begin{cases} f(t-a)\quad&; {t>a}\\ 0 \quad&;{t<a} \end{cases}, then \mathcal{L}\{g(t)\}=e^{-as}F(s).

Change of Scale

If f(t) is a function and F(s) is its Laplace transform, and c is a constant, then \mathcal{L}\{f(at)\}=\frac{1}{a}F(\frac{s}{a}).

Laplace Transform of Derivatives

1. If \mathcal{L}\{f(t)\}=F(s), then \mathcal{L}\{f'(t)\}=sF(s)-f(0), where f(t) is continuous for 0\le t\le N and of exponential order \gamma, and its derivative f'(t) is sectionally continuous for 0\le t\le N.
2. If f(t) fails to be continuous at t=0 but \lim\limits_{t\to0}f(t)=f(0^+), then \mathcal{L}\{f'(t)\}=sF(s)-f(0^+).
3. If f(t) fails to be continuous at t=a, then \mathcal{L}\{f'(t)\}=sF(s)-f(0)-e^{-as}\{f(a^+)-f(a^-)\}
4. If \mathcal{L}\{f(t)\}=F(s), then \mathcal{L}\{f^{(n)}(t)\}=s^nF(s)-s^{n-1}f(0)-s^{n-2}f'(0)-...-f^{(n-1)}(0), where f(t),\,f'(t),\,f''(t),\,...\,,f^{(n-1)}(t) are continuous for 0\le t\le N and of exponential order for t>N while f^{(n)}(t) is sectionally continuous for 0\le t\le N.

Laplace Transform of Integrals:

If \mathcal{L}\{f(t)\} =F(s), then \mathcal{L}\{\int\limits_0^t f(v)dv\}=\frac{F(s)}{s}.

Multiplication by t^n

If \mathcal{L}\{f(t)\} = F(s), then \mathcal{L}\{t^nf(t)\} = (-1)^n \frac{d^n}{ds^n}F(s), where \frac{d^n}{ds^n} denotes the n-th derivative.

Division by t

If \mathcal{L}\{f(t)\} = F(s), then \mathcal{L}\{\frac{f(t)}{t}\} = \int\limits_{s}^{\infty}f(v)dv.

Laplace Transform of Periodic functions

Let a function f(t) be periodic with period T>0, such that f(t+T)=f(t). Then \mathcal{L}\{f(t)\}=\frac{\int\limits_0^T e^{-st}f(t)dt}{1-e^{-sT}}.

Behavior of F(s) as s\to\infty

If \mathcal{L}\{f(t)\}=F(s), then \lim\limits_{s\to\infty}F(s)=0.

Initial Value Theorem

Let f(t) be a sectional continuous with Laplace transform F(s). Then \lim\limits_{s\to\infty}sF(s) = f(0^+), where the limit s\to\infty has to be taken in such a way that the real part of s, Re s \to\infty as well.

Final Value Theorem

Let f(t) be a sectional continuous with Laplace transform F(s). When f(\infty) =\lim\limits_{t\to\infty}f(t) exists, then \lim\limits_{s\to0}sF(s) = f(\infty), where the limit s\to0 has to be taken in such a way that the real part of s, Re(s) \to0 as well.

Convolution Theorem for Laplace Transform

Let f(t) and g(t) be piecewise continuous and of exponential order \gamma with their Laplace transforms F(s)=\mathcal{L}\{f(t)\} and G(s)=\mathcal{L}\{g(t)\}, respectively. Then \mathcal{L}\{f*g\} exists for Re(s)>\gamma and \mathcal{L}\{f*g\}=F(s)\cdot G(s).

Inverse Laplace Transformation:

If the Laplace transform of a function f(t) is F(s), i.e., if \mathcal{L}\{f(t)\}=F(s), then f(t) is called the inverse Laplace transform of F(s), and we write it symbolically as f(t)=\mathcal{L}^{-1}\{F(s)\} where \mathcal{L^{-1}} is called the inverse Laplace transformation operator.

• Example: Find the inverse Laplace transform of F(s)=\frac{1}{s+3}.
  We know that \mathcal{L}\{e^{at}\}=\frac{1}{s-a}, s>a. Then, \mathcal{L^{-1}}\{\frac{1}{s-a}\}=e^{at}. Similarly, \mathcal{L^{-1}}\{\frac{1}{s+3}\}=e^{-3t}.

Bilateral Laplace Transform

The bilateral Laplace transform involves the values of a function for both t<0 and t\ge0. This means that the bilateral Laplace transform is well-suited for non-causal signals and functions. If f(t) is the function and F(s) is its bilateral Laplace transform, then F(s) is given by:

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

Applications of Laplace Transform

Various applications of the Laplace transform include:

• The Laplace transform can be used to find the transfer function of linear time-invariant continuous-time systems: In linear time-invariant continuous-time systems (LTI systems), h(t) is the impulse response. The system function, or transfer function, H(s), of the LTI system is the Laplace transform of h(t).

• Laplace transform can be used to solve differential equation problems, including initial value problems. In an initial value problem, the solution to a differential equation is determined by the initial conditions of the system, such as the initial values of the function and its derivatives.

• Let H(s) be the transfer function of a causal LTI system described by the following differential equation:
  a_n \frac{d^n y(t)}{dt^n} + a_{n-1} \frac{d^{n-1} y(t)}{dt^{n-1}} + \cdots + a_1 \frac{dy(t)}{dt} + a_0 y(t) = b_m \frac{d^m x(t)}{dt^m} + b_{m-1} \frac{d^{m-1} x(t)}{dt^{m-1}} + \cdots + b_1 \frac{dx(t)}{dt} + b_0 x(t) satisfying the following condition of initial rest,
  y(0)=y'(0)=y''(0)=...=y^{m-2}(0)=y^{m-1}(0)=0 Then the system is stable if and only if the poles of H(s) lie in the left half-plane Re(s)<0.

• Laplace transform can be applied to analyze electrical circuits, simplifying the process of solving circuits with capacitors, inductors, and resistors by converting the time-domain equations into s-domain equations.

• Laplace transform is used in probability theory to find the distribution of sums of random variables and to solve problems related to stochastic processes. For example: Transforming Probability Density Functions (PDFs). The Laplace transform can be used to transform the probability density function (PDF) of a random variable. For a non-negative random variable X with PDF f_X(x), the Laplace transform is
  \mathcal{L}\{f_X(x)\} = F(s) = \int_0^\infty e^{-sx} f_X(x) \, dx.

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