Question

Calculate the Laplace transform of \(f(t)=\frac{\sin t}{t}\). (Hint: First compute \(\left.\frac{d}{d s} \mathcal{L}[f](s)_{+}\right)\)

Short answer

The Laplace transform of \(f(t)=\frac{\sin t}{t}\) is related to differentiation and simplification, involving limits for convergence.

Step-by-step solution

Understand the problem

We need to calculate the Laplace transform of a specific function, which is given as \(f(t) = \frac{\sin t}{t}\). This function is known as a sinc function, to solve this problem, it's useful to compute the derivative of the Laplace transform with respect to the parameter \(s\) first, as hinted.

Find the Laplace Transform of a related function

Consider the Laplace Transform of \(g(t) = \sin(t)\), which is the integral \( \mathcal{L}[\sin(t)] = \int_0^\infty e^{-st}\sin(t) \, dt \). This is \( \frac{1}{s^2 + 1} \). However, our function is \( \frac{\sin(t)}{t} \).

Use the Heaviside expansion theorem and related formulas

To solve \(f(t) = \dfrac{\sin t}{t}\) specifically, start by realizing that this is related to the derivative of \( \frac{\sin(t)}{t} \). Use the fact: \[ \frac{d}{ds} \mathcal{L}[rac{df(t)}{dt}] = -\mathcal{L}[f(t)] \], and the fact that if \(L(\sin t) = \frac{1}{s^2+1}\), the sinc function's Laplace transform involves differentiating something simpler.

Compute the derivative

Use the result \(\mathcal{L}[u(t)]=\frac{1}{s}\) implies \(\mathcal{L}[t^n]=\frac{n!}{s^{n+1}}\) and relate it using complex inversion formula. Compute \(\mathcal{L}(\frac{d}{ds}\sin(t))\) taking derivative inside the Laplace problem bounds.

Conclude the Laplace Transform

Finally apply the inverse Laplace theorem and related deduction on substitution to find that the Laplace transform of \( f(t)=\frac{\sin t}{t} \), noting the differentiation trick is \(-\frac{1}{s} e^{-s}\) and manage limits for convergence. Here address special cases about exponential factors.

Simplify and evaluate

Given the transformations and differentiations, solve using \( L(t e^{-at}) \to \text{sin substitution}\) for convergent series to assess function directly. The solution checks with the exponential reach zeroing out ase local step overlap.

Key concepts

Sinc Function

The sinc function, denoted as \( \text{sinc}(t) \), is a mathematical function often encountered in signal processing and other areas of physics and engineering. It is defined as \( \text{sinc}(t) = \frac{\sin(t)}{t} \) for \( t eq 0 \) and \( \text{sinc}(0) = 1 \). This function exhibits a peak at \( t = 0 \), where it takes the value of 1, and it oscillates as \( t \) increases or decreases. The unique property of the sinc function is that its height decreases with increasing \( t \), which dampens the oscillations over a larger domain.
A practical application of the sinc function is in the field of digital signal processing, where it plays a vital role in reconstruction of continuous signals from discrete ones through the concept of interpolation. The sinc function is fundamental in approximating and smoothing data.
The challenge in analyzing functions like \( \text{sinc}(t) \) is managing the computations tied to these oscillations and understanding their transformations, such as with the Laplace transform.

Derivative of Laplace Transform

The derivative of the Laplace Transform with respect to a parameter, typically \( s \), is an advanced technique used to simplify complex transforms by leveraging their derivatives. If you have a function \( f(t) \), and you find the Laplace Transform \( \mathcal{L}[f(t)] \), sometimes calculating the derivative \( \frac{d}{ds}\mathcal{L}[f(t)] \) can simplify the problem.
This concept applies particularly when the direct Laplace transform is not straightforward or when the function expresses some unique attributes through differentiation. For instance, if \( f(t) \) is \( \text{sinc}(t) \), the derivative method allows you to avoid direct integration, which could be cumbersome or intractable.
To use this method, first compute the Laplace Transform of a related, simpler function, and then differentiate that result with respect to \( s \). This requires understanding the derivative in the context of the transform and manipulating it to reflect the behavior of the function.

Heaviside Expansion Theorem

The Heaviside expansion theorem is a valuable tool in solving differential equations and analyzing complex systems. It helps expand functions in terms of simpler, easily manageable fractions involving the Laplace Transform. This theorem is essentially a method to break down complicated transforms into sums of simpler, more familiar partial fractions.
Applying the Heaviside expansion theorem involves these steps:

• Identifying the poles of the Laplace-transformed function, where the denominator goes to zero, as these reveal the function's roots.
• Expressing the transformed function as a sum of terms, where each term corresponds to a single pole; these simpler expressions can then be easily inverted back to time domain functions.

By doing so, you manage the complexity of inverse transforms, making it easier to find the final solution in the time domain. This is particularly useful when dealing with piecewise functions or functions with distinct discontinuities.

Convergence of Laplace Series

The convergence of Laplace series is crucial in ensuring the Laplace Transform correctly represents a function over its domain. Convergence refers to the tendency of a series or sequence to approach a specific value as more terms are added. When applying the Laplace transform, ensuring convergence means that the resulting transformed function accurately models the original function without diverging.
Several factors ensure convergence in a Laplace series:

• The function involved should be piecewise continuous over the given interval, ensuring it is well-behaved and suitable for transformation.
• Poles of the Laplace transformed function must lie within certain regions of convergence, specifically within the half-plane defined by the real part of \( s \).

Convergence is essential because it dictates the series' ability to reconstruct the original function when applying the inverse Laplace Transform. Managing convergence is, therefore, critical in both the analysis and synthesis of functions using this powerful mathematical tool.