Question

Prove that if \(f:[0, \infty) \rightarrow \mathbb{C}\) is piecewise continuous and \(p\)-periodic \((p>0)\), then $$ \mathcal{L}[f](s)=\frac{1}{1-e^{-p s}} \int_{0}^{p} e^{-s t} f(t) d t, \quad s>0 . $$

Short answer

The proof involves showing that the Laplace transform of a \(p\)-periodic function reduces via geometric series to the given expression.

Step-by-step solution

Understand the Laplace Transform Definition

The Laplace transform of a function \(f\) is defined as \(\mathcal{L}[f](s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\). Our task is to show that for a \(p\)-periodic function, this can be expressed in a special form.

Express Periodic Function

Since \(f\) is \(p\)-periodic, we have \(f(t+p) = f(t)\) for all \(t\). This allows us to express the Laplace transform in terms of one period, integrated repeatedly.

Break Down the Integral

Substitute the periodicity into the Laplace transform: \(\int_{0}^{\infty} e^{-st} f(t) \, dt = \sum_{n=0}^{\infty} \int_{np}^{(n+1)p} e^{-st} f(t) \, dt\). Here, we segment the infinite integral into intervals of length \(p\).

Change of Variables in Integral

For each segment \([np, (n+1)p]\), perform a change of variables \(u = t - np\), which simplifies each segment to \(\int_{0}^{p} e^{-s(u+np)} f(u) \, du\). Replace \(t\) with \(u + np\) inside the integral.

Factor Out the Exponential Decay Term

The integral \(\int_{0}^{p} e^{-s(u + np)} f(u) \, du\) becomes \(e^{-snp} \int_{0}^{p} e^{-su} f(u) \, du\). Therefore, the complete sum is \(\sum_{n=0}^{\infty} e^{-snp} \int_{0}^{p} e^{-su} f(u) \, du\).

Sum the Geometric Series

Recognize the sum \(\sum_{n=0}^{\infty} e^{-snp}\) as a geometric series with first term 1 and common ratio \(e^{-sp}\). This series sums to \(\frac{1}{1 - e^{-sp}}\), where the series converges for \(s > 0\).

Combine the Results

Substituting back into our expression for the Laplace transform results in \(\mathcal{L}[f](s) = \frac{1}{1 - e^{-sp}} \int_{0}^{p} e^{-su} f(u) \, du\). This proves the original statement.

Key concepts

Periodic Functions

Periodic functions are fundamental in understanding various mathematical concepts, especially when dealing with Laplace transforms. A periodic function is one that repeats its values at regular intervals, known as the period. In our exercise, we consider a function \( f \) which is \( p \)-periodic. This means that for every \( t \), we have \( f(t + p) = f(t) \).

This property allows us to evaluate the function over just one period and then apply that evaluation to the entire timeline.

• The Laplace transform can take advantage of this repetition by calculating the integral over one period and multiplying by its effect along the whole domain.
• The periodicity simplifies the Laplace transformation by reducing redundant calculations over repeating cycles.

Understanding and recognizing periodic functions can greatly simplify many calculations and analyses, which is especially useful when trying to prove integral equivalences as presented in this exercise.

Geometric Series

In mathematics, a geometric series is a sum of terms where each term after the first is found by multiplying the previous term by a constant. This constant is called the common ratio.

In the exercise, we identify the series \( \sum_{n=0}^{\infty} e^{-snp} \) as a geometric series, where \( e^{-sp} \) acts as the common ratio. A geometric series can be summed to a closed form, which drastically simplifies computations.

• The sum of an infinite geometric series with first term \( a \) and common ratio \( r \) (where \(|r| < 1\)) is given by \( \frac{a}{1-r} \).
• Recognizing and applying this formula is essential to moving from a complex integral sum to a more manageable algebraic equation.

By understanding geometric series, we see how repeated patterns in mathematical terms can become very simplified expressions.

Piecewise Continuity

A function is piecewise continuous if it is continuous within each piece of a divided domain. Essentially, despite possible discontinuities at certain points, each segment of the function is individually continuous. This is critical when working with Laplace transforms.

In the context of our exercise, we consider \( f \) to be piecewise continuous. This ensures that the integral of \( f \), which defines the Laplace transform, is well-behaved and computable.

• Piecewise continuity ensures that a function can be integrated part by part, removing concerns of undefined behavior during integration.
• It allows handling of more complex functions that might have breakpoints, making them feasible to use within the scope of Laplace transforms.

Understanding piecewise continuity enables us to reliably use and manipulate functions across discontinuities, which plays a vital role when working with the Laplace transform.

Change of Variables

The change of variables is a fundamental mathematical tool used to simplify integrals. By substituting a new variable, we can make the bounds simpler or otherwise alter the form of the integral to make it more computable.

In our exercise, we perform a change of variables for each segment of the integral. By using \( u = t - np \), we shift the bounds to \( [0, p] \), simplifying integration of the periodic function.

• This technique helps in managing and solving complex integrals by focusing on a more approachable segment of the function's domain.
• It also facilitates the simplification of calculations in periodic functions when connected to infinite series.

Recognizing when and how to apply a change of variables significantly enhances our ability to solve complex integrals, such as those required in applying the Laplace transform to periodic functions.