Question

Let \(f\) be a \(2 \pi\)-periodic piecewise continuous function satisfying $$ \int_{-\pi}^{\pi} f(x) d x=0 . $$ \(\operatorname{Set} g(x)=\int_{0}^{x} f(t) d t\). (a) Prove that \(g\) is \(2 \pi\)-periodic. (b) Let \(\sum_{n=-\infty}^{\infty} c_{n} e^{i n x}\) be the complex Fourier series of the function \(f\) and \(\sum_{n=-\infty}^{\infty} d_{n} e^{i n x}\) the complex Fourier series of the function \(g\). Prove that for all real \(x\) we have the equality $$ g(x)=\sum_{n=-\infty}^{\infty} d_{n} e^{i n x} $$ where \(d_{n}=\frac{c_{n}}{i n}\) for every integer \(n \neq 0\).

Short answer

(a) \( g(x) \) is \( 2\pi \)-periodic because \( g(x+2\pi) = g(x) \). (b) \( d_n = \frac{c_n}{i n} \) for integers \( n \neq 0 \).

Step-by-step solution

Understanding the Periodicity

To prove that \( g(x) \) is \( 2\pi \)-periodic, note that \( f(x) \) is given as \( 2\pi \)-periodic. For \( g(x) = \int_0^x f(t) \, dt \), consider \( g(x + 2\pi) \): \[ g(x + 2\pi) = \int_0^{x+2\pi} f(t) \, dt = \int_0^{2\pi} f(t) \, dt + \int_{2\pi}^x f(t) \, dt \]Since \( \int_0^{2\pi} f(t) \, dt = 0 \), we have:\[ g(x + 2\pi) = \int_0^x f(t) \, dt = g(x) \]Thus, \( g(x) \) is \( 2\pi \)-periodic.

Understanding the Fourier Series Expansion

Given the complex Fourier series for \( f \) as \( \sum_{n=-\infty}^{\infty} c_n e^{i n x} \), and the series for \( g \) as \( \sum_{n=-\infty}^{\infty} d_n e^{i n x} \), we need to show the relationship between \( c_n \) and \( d_n \) where \( d_n = \frac{c_n}{i n} \) for \( n eq 0 \).

Relating g's Fourier Coefficients to f's

Utilize the integral property and initial condition to deduce \( d_n = \frac{c_n}{i n} \). Since \( g'(x) = f(x) \), and\[ g(x) = \int_0^x f(t) \, dt \Rightarrow g'(x) = f(x) \]Apply the Fourier series relationship for differentiation: if \( f(x) \) has coefficients \( c_n \), then differentiating \( g(x) \) results in coefficients of form \( i n d_n = c_n \) for \( n eq 0 \). Rearranging gives:\[ d_n = \frac{c_n}{i n} \].For \( n=0 \), since \( \int_{-\pi}^{\pi} f(x) \, dx = 0 \), \( c_0 = 0 \), giving \( d_0 \) indeterminate, typically set to zero.

Key concepts

Periodicity

Periodicity is a key concept in mathematics that describes the repeating nature of a function over regular intervals. A function is said to be periodic if it repeats its values in regular intervals or periods. In our context, we'll focus on functions that are periodic over a period of \(2\pi\).

This is often expressed as \(f(x + T) = f(x)\) where \(T\) is the period. For the function \((g(x))\) under our exercise, it means that after every interval of \(2\pi\), the function essentially "starts over", giving it a periodic nature.

In the solution, we use the periodicity of \(f(x)\), a \(2\pi\)-periodic function, to prove that \(g(x)\) is also periodic with the same period. By performing an integral transformation and utilizing the fact that \(\int_{0}^{2\pi} f(t) \, dt = 0\), it was shown that shifting \(g(x)\) by \(2\pi\) results in the same function value, \(g(x + 2\pi) = g(x)\). This proves that \(g(x)\) maintains its periodic nature over \(2\pi\).

Understanding periodicity is crucial in analyzing signals and waves, and it's an important property in many branches of physics and engineering.

Piecewise Continuous Functions

A piecewise continuous function is a function composed of different "pieces" or segments, where each segment is continuous within its domain. This does not necessarily mean the function has to be continuous at the segment borders, but it cannot have infinite discontinuities. This means it can safely be integrated, which is crucial in many applications such as signal processing and system analysis.

In the context of our exercise, \(f(x)\) is a \(2\pi\)-periodic piecewise continuous function. The task especially focuses on utilizing its integral properties, ensured by its piecewise continuity, to determine the periodicity of \(g(x)\) and to analyze its Fourier series. The integral ensures that we account for changes across these different sections of \(f(x)\), seamlessly giving us usable outcomes like the complex Fourier series.

So, although the function can change its shape at different intervals, the integral can be taken from the beginning to the end of the interval without issues, as the discontinuities are not infinite. This is foundational in performing mathematical transformations such as the Fourier series.

Complex Fourier Series

The complex Fourier series is a representation of a periodic function as a sum of complex exponentials. Unlike the standard Fourier series that utilizes sines and cosines, the complex form makes use of Euler's formula, \(e^{inx} = \cos(nx) + i\sin(nx)\), which provides significant mathematical benefits in simplifying expressions and calculations.

This representation applies well to our function \(f(x)\) with coefficients \(c_n\) and \(g(x)\) with coefficients \(d_n\). In our exercise, the focus is on deriving the relationship between these sets of coefficients. Specifically, we used the property that the derivative of the Fourier series yields another series, indicating that \(g'(x) = f(x)\). This leads to the relationship \(in d_n = c_n\) for \(n eq 0\). Solving gives us \(d_n = \frac{c_n}{i n}\).

For \(n = 0\), since the integral evaluated to zero, both \(c_0\) and \(d_0\) vanish, ensuring no constant term. Understanding this relationship helps in analyzing and synthesizing signals, a key application of Fourier analysis, particularly in electrical engineering and acoustic studies.