Problem 11 - Advanced Engineering Mathematics

Question

(a) Adding \(c_{n}=\frac{1}{2}\left(a_{n}-i b_{n}\right)\) and \(c_{-n}=\frac{1}{2}\left(a_{n}+i b_{n}\right)\) we get \(c_{n}+c_{-n}=a_{n} .\) Subtracting, we get \(c_{n}-c_{-n}=-i b_{n}\) Multiplying both sides by \(i\) we obtain \(i\left(c_{n}-c_{-n}\right)=b_{n}\) (b) From $$a_{n}=c_{n}+c_{-n}=(-1)^{n} \frac{\sinh \pi}{\pi}\left[\frac{1-i n}{n^{2}+1}+\frac{1+i n}{n^{2}+1}\right]=\frac{2(-1)^{n} \sinh \pi}{\pi\left(n^{2}+1\right)}, \quad n=0,1,2, \ldots$$ and $$b_{n}=i\left(c_{n}-c_{-n}\right)=i(-1)^{n} \frac{\sinh \pi}{\pi}\left[\frac{1-i n}{n^{2}+1}-\frac{1+i n}{n^{2}+1}\right]=i(-1)^{n} \frac{\sinh \pi}{\pi}\left[-\frac{2 i n}{n^{2}+1}\right]=\frac{2(-1)^{n} n \sinh \pi}{\pi\left(n^{2}+1\right)}$$ the Fourier series of \(f\) is $$f(x)=\frac{\sinh \pi}{\pi}+\frac{2 \sinh \pi}{\pi} \sum_{n=1}^{\infty}\left[\frac{(-1)^{n}}{n^{2}+1} \cos n x+\frac{n(-1)^{n}}{n^{2}+1} \sin n x\right]$$

Step-by-Step Solution

Answer

The Fourier series for \(f(x)\) is constructed using derived expressions for \(a_n\) and \(b_n\).

Step 1: Combine Expressions for c_n and c_-n

We are given expressions for \(c_n\) and \(c_{-n}\):\[c_{n} = \frac{1}{2}(a_n - i b_n), \c_{-n} = \frac{1}{2}(a_n + i b_n)\]Adding these together gives:\[c_n + c_{-n} = \frac{1}{2}(a_n - ib_n) + \frac{1}{2}(a_n + ib_n) = a_n\]

Step 2: Subtract Expressions for c_n and c_-n

Subtract the expression of \(c_{-n}\) from \(c_n\):\[c_n - c_{-n} = \frac{1}{2}(a_n - ib_n) - \frac{1}{2}(a_n + ib_n) = -ib_n\]Multiplying both sides by \(i\):\[i(c_n - c_{-n}) = b_n\]

Step 3: Evaluate Expression for a_n

The expression for \(a_n\) is given by:\[a_n = c_n + c_{-n} = (-1)^n \frac{\sinh \pi}{\pi}\left[\frac{1-in}{n^2+1} + \frac{1+in}{n^2+1}\right] = \frac{2(-1)^n \sinh \pi}{\pi(n^2+1)}\]

Step 4: Evaluate Expression for b_n

The expression for \(b_n\) is given by:\[b_n = i(c_n - c_{-n}) = i(-1)^n \frac{\sinh \pi}{\pi}\left[\frac{1-in}{n^2+1} - \frac{1+in}{n^2+1}\right] = \frac{2(-1)^n n \sinh \pi}{\pi(n^2+1)}\]

Step 5: Write the Fourier Series of f(x)

Using the expressions for \(a_n\) and \(b_n\), the Fourier series for \(f(x)\) is:\[f(x)=\frac{\sinh \pi}{\pi} + \frac{2 \sinh \pi}{\pi}\sum_{n=1}^{\infty}\left[\frac{(-1)^n}{n^2+1} \cos(nx) + \frac{n(-1)^n}{n^2+1} \sin(nx)\right]\]

Key Concepts

Sinh Function

The sinh function, or hyperbolic sine, is an important function when dealing with series and complex expressions. It is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). This function behaves similarly to the sine function from trigonometry, but is based in the realm of hyperbolic geometry.

Just as the sine function is part of the fundamentals of trigonometry, the sinh function is fundamental in hyperbolic functions, which are critical in advanced mathematics and physics. These hyperbolic functions, like \( \sinh(x) \), are used extensively in areas such as high-speed motion, electrical engineering, and in solving differential equations.

• The sinh function grows exponentially as \( x \) increases or decreases, unlike the sinusoidal growth of the sine function.
• Similar to trigonometric identities, there are hyperbolic identities, such as \( \cosh^2(x) - \sinh^2(x) = 1 \).
• In the context of series expansion, \( \sinh(x) \) can be expressed as an infinite series: \( \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \ldots \), showcasing its relationship with exponential functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any value of the variable involved. They play a significant role in simplifying expressions and solving equations. Some common trigonometric identities include:

• Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle sum and difference identities: \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)
• Double angle identities: \( \sin(2x) = 2\sin(x)\cos(x) \), \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

In the context of Fourier series, trigonometric identities are pivotal. They help break down complex waveforms into their sine and cosine components, which are more manageable mathematically. In particular, the expressions for \(a_n\) and \(b_n\) in the given problem can be rewritten using these identities, which ease the manipulation of trigonometric expressions in series expansions.

These identities bridge the gap between simple trigonometric concepts and comprehensive analysis, helping to solve various problems in physics and engineering.

Series Expansion

Series expansion is a mathematical tool used to represent functions as the sum of terms in an infinite sequence, often simplifying the manipulation and solution of complex problems. This method is especially useful for approximating functions, performing complex calculations, and solving differential equations.

An example of a series expansion is the Taylor series, which represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point:

• Taylor series of \(f(x)\) about a point \(a\): \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \]
• Power series: A series of the form \( \sum_{n=0}^{\infty} a_n x^n \).
• Fourier series: Expressing a function as a sum of sine and cosine terms, as seen in the problem above.

The utility of series expansions lies in their ability to approximate even very complicated functions using simpler polynomial forms. The Fourier series in particular, used in the solution, is vital for representing periodic functions by breaking them down into their constituent sinusoidal components. This is critical for analyzing complex waveforms in physics and engineering contexts, enabling engineers and scientists to derive meaningful insights from otherwise intractable functions.