Problem 8 - Advanced Engineering Mathematics

Question

Adding the series for \(e^{z}\) and \(e^{-z}\) gives \(\cosh z=\frac{1}{2}\left(e^{z}+e^{-z}\right)=\sum_{k=0}^{\infty} \frac{z^{2 k}}{(2 k) !}\) where \(R=\infty\)

Step-by-Step Solution

Answer

The series describes \( \cosh z \), with even powers of \( z \).

Step 1: Understanding the Exponential Series

The functions, \( e^z \) and \( e^{-z} \), can be expressed as their corresponding series expansions: \( e^z = \sum_{k=0}^{\infty} \frac{z^k}{k!} \) and \( e^{-z} = \sum_{k=0}^{\infty} \frac{(-z)^k}{k!} \). Let's analyze these expressions step by step.

Step 2: Deriving the Expression for \( e^z + e^{-z} \)

We add the series for \( e^z \) and \( e^{-z} \):\[e^z + e^{-z} = \sum_{k=0}^{\infty} \frac{z^k}{k!} + \sum_{k=0}^{\infty} \frac{(-z)^k}{k!}.\]

Step 3: Analyzing the Resulting Terms in the Summation

Separate the terms in the summation by their parity (i.e., even or odd powers of \( z \)), because they affect the cancellation or addition in terms:\[= \left( \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!} \right) + \left( \sum_{n=0}^{\infty} \frac{z^{2n+1}}{(2n+1)!} \right) + \left( \sum_{n=0}^{\infty} \frac{(-z)^{2n}}{(2n)!} \right) + \left( \sum_{n=0}^{\infty} \frac{(-z)^{2n+1}}{(2n+1)!} \right).\]

Step 4: Cancellation of Odd Terms

The odd terms in the series cancel each other out due to the alternating signs in \( e^z \) and \( e^{-z} \):\[e^z + e^{-z} = 2 \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}. \]

Step 5: Identifying the Expression for \( \cosh z \)

The function \( \cosh z \) is defined as:\[\cosh z = \frac{1}{2} (e^z + e^{-z}) = \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}\] noting that the \( \frac{1}{2} \) factor cancels with the multiplied 2.

Key Concepts

Exponential Series

The exponential series is a way to represent the exponential function using an infinite sum of terms. This series expands the mathematical expression of the exponential function, which is frequently encountered in mathematics due to its applications in many fields such as physics, engineering, and statistics. The exponential function, often denoted as \(e^z\), can be expanded into a series:

• \(e^z = \sum_{k=0}^{\infty} \frac{z^k}{k!}\)
• \(e^{-z} = \sum_{k=0}^{\infty} \frac{(-z)^k}{k!}\)

This representation allows us to understand how the exponential function can be broken down into an infinite sum of powers of \(z\). Each term in the series is derived from raising \(z\) to the power \(k\), then dividing by the factorial of \(k\) (i.e., \(k!\)). This infinite series includes both positive and negative powers of \(z\), giving a comprehensive view of the function's behavior over different values of \(z\).
The series expansion is particularly useful for approximating the value of the exponential function and helps simplify complex calculations involving exponentials.

Cosh Function

The hyperbolic cosine function, known as the \(\cosh\) function, is an important hyperbolic function that mirrors many properties of the regular cosine function but differs in its mathematical construction. It is defined for all real numbers \(z\), and its relation to the exponential function is as follows:

• \(\cosh z = \frac{1}{2}(e^z + e^{-z})\)

This equation highlights how the \(\cosh\) function combines the exponential function with its inverse, \(e^{-z}\). It's this addition that leads to the excellent properties of the \(\cosh\) function, making it an "even" function.
This behavior can be observed in its graph, which is symmetric about the y-axis, reinforcing its status as an even function. Applications of \(\cosh z\) are pervasive across many areas of science and engineering, such as the description of hyperbolic curves and in analyses involving wave motion.

Series Expansion

Series expansion refers to the method of expressing more complex functions as an infinite sum of simpler terms, often powers of a variable. This is extremely useful in calculus and other advanced branches of mathematics because it allows one to analyze functions in a more digestible form.The expression of \(e^z + e^{-z}\) as a series allows us to identify the hyperbolic cosine function, \(\cosh z\), in a more explicit form:

• \(e^z + e^{-z} = 2 \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}\)

Here, the series only includes even powers of \(z\) due to the specific behavior of the exponential functions involved. By expanding complex expressions into series, we can simplify otherwise challenging mathematical problems, facilitating easier computation and understanding.
The technique of series expansion is also foundational in developing approximations for other functions, particularly in contexts where a graphical depiction or numerical calculation is needed.

Even and Odd Functions

Understanding even and odd functions is crucial in mathematics, especially when dealing with symmetry in function graphs. An even function, by definition, satisfies \(f(-x) = f(x)\) for all \(x\). In contrast, an odd function meets the condition \(f(-x) = -f(x)\).
The hyperbolic cosine function is a classic example of an even function because \(\cosh(-z) = \cosh(z)\). This symmetry induces specific consequences in its series expansion:

• For even functions like \(\cosh z\), only the terms involving even powers of \(z\) remain after simplification.
• The formula \(\cosh z = \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}\) reflects this evenness, showcasing the exclusion of any odd-powered terms that would otherwise cancel out.

This exclusion of odd terms is a result of the symmetry in even functions, leading to more efficient computation and analysis techniques when dealing with such functions. Good grasp on even and odd functions assists in understanding the behavior and properties of different mathematical models.