Problem 55 - Advanced Engineering Mathematics

Question

Suppose \(f(x)=e^{x}, 0

Step-by-Step Solution

Answer

Yes, the averaged series is the full Fourier series of the function.

Step 1: Understand the Problem

We are given a function \( f(x) = e^{x} \) defined on the interval \((0, \pi)\). We need to examine the behavior after it is expanded into cosine and sine series separately, averaged, and determine if this results in a full Fourier series.

Step 2: Expansion into Cosine and Sine Series

Expand \( f(x) = e^{x} \) into a cosine series on \(0 < x < \pi\) which will include only cosine terms since cosine functions account for even extensions. Similarly, expand \( f(x) = e^{x} \) into a sine series on \(0 < x < \pi\), which will include only sine terms since sine functions account for odd extensions.

Step 3: Averaging the Series

When you average the two expansions (cosine and sine series), you take \(\frac{1}{2}(\text{cosine series} + \text{sine series})\). This averaging means you are calculating a mix of even and odd extensions.

Step 4: Interpretation of the Averaging

Consider the effect of averaging these series on the interval \((-\pi, \pi)\). On \((0, \pi)\), the cosine and sine series individually represent \( f(x) = e^{x} \). The averaging implies a symmetrical extension that accounts for both even and odd functions, effectively capturing the full behavior across \((-\pi, \pi)\).

Step 5: Conclusion on Full Fourier Series

By averaging the cosine and sine series, we essentially model both the even and odd symmetry properties of \( f(x) = e^{x} \). The resulting series corresponds to the full Fourier series of \( f(x) \) over the interval \((-\pi, \pi)\).

Key Concepts

Cosine Series

In the realm of Fourier series, the cosine series plays a crucial role due to its focus on even functions. A function is even if it satisfies the equation \(f(x) = f(-x)\). Think of it like a mirror reflection along the y-axis. For instance, \(\cos(x)\) is even because \(\cos(x) = \cos(-x)\).

When we expand a function into a cosine series, we are assuming that it behaves like an even function outside its original interval. This means we are focusing on the portion of the Fourier series that uses cosine terms to approximate the function. The cosine series does not consider any sine terms because sine functions would reflect as odd around the origin.

• Cosine series are represented by a sum of cosine terms: \(a_0 + a_1 \cos(x) + a_2 \cos(2x) + ...\).
• Good for even functions or functions defined in a symmetric interval around zero.
• Essential in representing symmetrical extensions of functions.

Understanding the cosine series helps in covering half of the story when dealing with Fourier series, particularly with functions that naturally extend in an even manner. It guarantees that the information of the original interval is mirrored accurately across the entire interval involved in the extension.

Sine Series

In contrast to the cosine series, the sine series is vital for representing odd functions. An odd function satisfies \(f(x) = -f(-x)\). This represents a kind of rotational symmetry around the origin. For example, \(\sin(x)\) is an odd function because \(\sin(-x) = -\sin(x)\). When examining full Fourier series, incorporating the sine components is crucial for accounting for the oddities (pun intended) of the function.

Expanding a function into a sine series indicates extending the function in a way that causes a change of sign or rotational symmetry at the origin, potentially doubling the function's range in terms of symmetry. This is because sine series consider terms that inherently switch signs halfway through their period.

• Sine series consist of sine terms: \(b_1 \sin(x) + b_2 \sin(2x) + ...\).
• Best for functions with "odd" properties, offset from the x-axis.
• Key in designing odd extensions of periodic functions within Fourier analysis.

The inclusion of the sine series ensures that the full picture of how a function could behave within its extended interval is considered alongside the representation of the even characteristics obtained via the cosine series.

Even and Odd Functions

In mathematics, understanding the symmetry of functions is vital for Fourier analysis. Even and odd functions are central concepts that facilitate the creation of Fourier series. Even functions, denoted by \(f(x) = f(-x)\), reflect perfectly across the y-axis. In contrast, odd functions, denoted by \(f(x) = -f(-x)\), have rotational symmetry about the origin. These properties help us construct more efficient representations of complex functions using simple sine and cosine waves.

• Even functions are built using cosine terms in Fourier series, emphasizing symmetry without sign change.
• Odd functions use sine terms, reflecting symmetry with a sign change, crucial for representing alternating behaviors.
• Combining both gives a comprehensive picture through the full Fourier series, capturing the entire interval's behavior.

The interplay between even and odd functions in Fourier series enables us to piece together a complete mathematical puzzle, enhancing our ability to model periodic phenomena. By understanding these foundational concepts, students can better appreciate how the averaging of functions' even and odd components leads to full and accurate function representations across specified intervals.