Question

In Problems \(5.1\) to \(5.8\), define each function by the formulas given but on the interval \((-l, l)\). [That is, replace \(\pm \pi\) by \(\pm l\) and \(\pm \pi / 2\) by \(\pm 1 / 2\).) Expand each function in a sine-cosine Fourier series and in a complex exponential Fourier series.

Short answer

Fourier sine-cosine series: \[ f(x) = a_0 + \sum_{n=1}^{\infty} [a_n cos( \frac{n \pi x}{l} ) + b_n sin ( \frac{n \pi x}{l} )] \]. Complex exponential Fourier series: \[ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{\frac{in\pi x}{l}} \].

Step-by-step solution

Define the Function on the Interval

Define each function by using the properties given. Replace \[ \pm \pi \] with \[ \pm l \] and \[- \pi / 2 \] with \[ - l / 2 \].

Determine Fourier Coefficients (Sine-Cosine Series)

Calculate the Fourier coefficients for the sine-cosine series: \[ a_0, a_n, \] and \[ b_n \]. Use the formulas: \[ a_0 = \frac{1}{l} \int_{-l}^{l} f(x)\,dx \, \ a_n = \frac{1}{l} \int_{-l}^{l} f(x)\cos(\frac{n\pi x}{l})dx \, \ b_n = \frac{1}{l} \int_{-l}^{l} f(x) sin( \frac{n\pi x}{l} ) dx \].

Write the Sine-Cosine Fourier Series

Formulate the sine-cosine Fourier series: \[ f(x) = a_0 + \sum_{n=1}^{\infty} [a_n cos( \frac{n \pi x}{l} ) + b_n sin ( \frac{n \pi x}{l} )] \].

Determine Fourier Coefficients (Complex Exponential Series)

Calculate the Fourier coefficients for the complex exponential Fourier series using: \[ c_n = \frac{1}{2l} \int_{-l}^{l} f(x)e^{-\frac{in\pi x}{l}}dx \].

Write the Complex Exponential Fourier Series

Formulate the complex exponential Fourier series: \[ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{\frac{in\pi x}{l}} \].

Key concepts

Sine-Cosine Series

The sine-cosine series is a way to express a function as an infinite sum of sine and cosine terms. This approach uses real-valued sinusoidal functions. Start by defining the function on a specified interval, replacing common boundaries like \([- \pi, \pi]\) with \([-l, l]\). Next, compute the Fourier coefficients which are essential for constructing the series.

• The coefficient \(a_0\) is calculated using the formula: \[ a_0 = \frac{1}{l} \int_{-l}^{l}f(x)dx \] This represents the average value of the function over the interval.
• The coefficients \(a_n\) and \(b_n\) are found using: \[ a_n = \frac{1}{l} \int_{-l}^{l}f(x)\cos\(\frac{n\pi x}{l}\)dx, \ b_n = \frac{1}{l} \int_{-l}^{l}f(x)\sin\(\frac{n\pi x}{l}\)dx \] Here, \(a_n\) captures the cosine parts and \(b_n\) captures the sine parts of the function.

Lastly, combine these coefficients to form the sine-cosine Fourier series: \[ f(x) = a_0 + \sum_{n=1}^{\infty}[a_n\cos(\frac{n\pi x}{l}) + b_n\sin(\frac{n\pi x}{l})] \] This series approximates the function by adding up an infinite number of sine and cosine terms.

Complex Exponential Series

The complex exponential series expresses a function as a sum of exponential functions involving complex numbers. Introduce the interval and replace usual limits like \([-\pi, \pi]\) with \([-l, l]\).

The Fourier coefficients, denoted as \(c_n\), are crucial for constructing this series. These are calculated using the formula: \[ c_n = \frac{1}{2l}\int_{-l}^{l}f(x)e^{-\frac{in\pi x}{l}}dx \] Here, \(i\) is the imaginary unit and helps in accommodating complex numbers.

Once you have the coefficients, form the series as: \[ f(x) = \sum_{n=-\infty}^{\infty}c_n e^{\frac{in\pi x}{l}} \] This representation is advantageous because it uses the compact exponential form and can handle both periodic and aperiodic signals efficiently.

Compared to the sine-cosine series, the complex exponential series can simplify calculations, especially when dealing with signals and systems in electrical engineering and physics.

Fourier Coefficients

Fourier coefficients are the building blocks of Fourier series, representing the weights of different frequency components.

• For the sine-cosine series, the coefficients tell us how much of each sine and cosine function is needed to reconstruct the original function. \(a_0\) is the average value, while \(a_n\) and \(b_n\) capture the amplitudes of the cosine and sine components respectively.
• For the complex exponential series, \(c_n\) coefficients capture both amplitude and phase information of each complex exponential term. They utilize both real and imaginary parts to account fully for the signal's properties.

Calculating these coefficients involves integration over the given interval. Precision in determining these values is crucial for an accurate reconstruction of the original function.

Understanding Fourier coefficients' role can help one grasp how different sinusoidal or exponential components come together to represent complex periodic or even aperiodic functions effectively.