Question

Parts (a) and (b), you are given in each case one period of a function. Sketch several periods of the function and expand it in a sine-cosine Fourier series, and in a complex exponential Fourier series. (a) $f(x)=e^x,-\pi <x<\pi$ (b) $f(x)=e^x,0<x<2\pi$.

Short answer

For (a), the Fourier coefficients are determined over $-\pi <x<\pi$. For (b), the interval is $0<x<2\pi$. The series expansions involve computing integrals to find coefficients.

Step-by-step solution

- Understand the Problem

We need to sketch several periods of the given functions and expand them in both sine-cosine Fourier series and complex exponential Fourier series. The functions given are periodic but defined over different intervals.

- Define the Periodic Extension

For each function, extend it periodically beyond the given interval to see how it repeats.

Part (a) - Sketch the Function

For the function $f(x)=e^x$ in the interval $-\pi <x<\pi$, periodically extend this by repeating the pattern every $2\pi$. Sketch several periods of this extension.

Part (a) - Fourier Series: Determine Coefficients

Use the Fourier series formulas for a periodic function with period $2\pi$. The coefficients are given by: $$a_0=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{e}^{x}dx$$$$a_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }{e}^x\cos (nx)dx$$$$b_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }{e}^x\sin (nx)dx$$

Part (a) - Compute Coefficients

Calculate the integrals for the Fourier coefficients. $$a_0=\frac{2\sinh (\pi )}{\pi }$$$$a_n=\frac{2(\sinh (\pi )\cos (n\pi )-\cosh (\pi )\sin (n\pi ))}{\pi (1+n^2)}$$$$b_n=\frac{2(\sinh (\pi )\sin (nx)+\cosh (\pi )\cos (nx))}{\pi (1+n^2)}$$

Part (a) - Write Sine-Cosine Fourier Series

The function can be written as$$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos (nx)+b_n\sin (nx))$$

Part (a) - Complex Exponential Fourier Series

The complex exponential Fourier series is given by:$$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{inx}$$where \ is $$c_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{e}^x{e}^{-inx}dx$$

Part (b) - Sketch the Function

For the function $f(x)=e^x$ in the interval $0<x<2\pi$, periodically extend this by repeating the pattern every $2\pi$. Sketch several periods of this extension.

Part (b) - Fourier Series: Determine Coefficients

Use the Fourier series formulas for a periodic function with period $2\pi$. The coefficients are given by: $$a_0=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{x}dx$$$$a_n=\frac{1}{\pi }{\int }_0^{2\pi }{e}^x\cos (nx)dx$$$$b_n=\frac{1}{\pi }{\int }_0^{2\pi }{e}^x\sin (nx)dx$$

Part (b) - Compute Coefficients

Calculate the integrals for the Fourier coefficients. For part (b), these integrals need to be adjusted to the interval [0, 2π]:$$a_0=\frac{e^{2\pi }-1}{2(\pi )}$$$$a_n=\frac{2(\cosh (2\pi )\cos (n2\pi )-\sinh (2\pi )\sin (n2\pi ))}{\pi (1+n^2)}$$$$b_n=\frac{2(\cosh (2\pi )\sin (nx)+\sinh (2\pi )\cos (nx))}{\pi (1+n^2)}$$

Part (b) - Write Sine-Cosine Fourier Series

The function can be written as$$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos (nx)+b_n\sin (nx))$$

Part (b) - Complex Exponential Fourier Series

The complex exponential Fourier series is given by:$$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{inx}$$where \ is $$c_n=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^x{e}^{-inx}dx$$

Key concepts

Periodic Functions

A periodic function repeats its values in regular intervals or periods. The simplest example is the sine function, which repeats every $2\pi$ radians. When dealing with periodic functions, it is crucial to determine the interval over which they repeat. This interval is called the period. For example, a function $f(x)$ with a period $2\pi$ satisfies $f(x+2\pi )=f(x)$ for all $x$. For the exercise given, we need to treat the exponential functions $e^x$ as periodic by extending them over the specified intervals.

Sine-Cosine Fourier Series

Fourier series allows us to express a periodic function as a sum of sine and cosine terms. This is useful because sine and cosine functions are orthogonal, making calculations straightforward.
The sine-cosine Fourier series of a periodic function $f(x)$ with period $2\pi$ is given by
$$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\cos (nx)+b_n\sin (nx))$$The coefficients $a_n$ and $b_n$ are calculated using the integrals:$$\text{[}\begin{array}{rlrl}{a}_0& =\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)dx{a}_n& =\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (nx)dx{b}_n& =\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx\end{array}$$\]
These formulas allow us to break down any periodic function into a series of sines and cosines.

Complex Exponential Fourier Series

Another way to express a periodic function is through the complex exponential Fourier series. Using Euler's formula ($e^{ix}=\cos (x)+i\sin (x)$), we can represent sine and cosine functions as complex exponentials. This is often more compact and elegant.
The complex exponential Fourier series for a function $f(x)$ with period $2\pi$ is given by
$$f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{inx}\text{ewline}$$
Where the coefficients $c_n$ are calculated using the integral:ewline$$c_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)e^{-inx}dx$$

• Compared to the sine-cosine series, this series can often simplify the algebra involved in working with periodic functions.
• The coefficients $c_n$ may be complex numbers, but the series itself will still represent a real-valued function if the original function is real.

Fourier Coefficients Calculation

The Fourier coefficients are the key to the Fourier series, as they determine the amplitude of each sine and cosine wave (or exponential term). For the sine-cosine series:

• Calculate $a_0$: the average value of the function over one period. This is done by integrating the function over one period and dividing by the length of the period.
• Calculate $a_n$ and $b_n$: These coefficients give the amplitudes of the cosine and sine terms, respectively.
• Integrate $f(x)\cos (nx)$ and $f(x)\sin (nx)$ over one period to find $a_n$ and $b_n$.

For the complex exponential series:

• Calculate $c_n$: Integrate $f(x)e^{-inx}$ over one period and divide by the period.

These coefficients allow us to construct the full Fourier series representation of the function, which can be used for analyzing or approximating the function.