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Introduction to Half-Range Expansions
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Today, we're introducing how Fourier series can be adjusted for functions defined only on a part of their domain, specifically between 0 and L. We call this technique 'Half-Range Expansions'.
What do you mean by 'half-range'? Is it just about using half of the series?
Great question! 'Half-range' refers to using the Fourier sine and cosine series for functions defined on 0 < x < L. Instead of extending the function to be periodic, we adapt it using odd and even extensions.
How do we choose between sine and cosine for these expansions?
Another excellent query! If we're dealing with odd functions that go to zero at the endpoints, we use half-range sine series. For even functions that maintain symmetry, we use half-range cosine series.
Can you give us an example of when we would use each?
Sure! For a temperature distribution on a rod fixed at both ends, we'd use sine series. But for a puddle of water exhibiting radial symmetry, cosine series would be suitable.
To summarize, half-range expansions let us tackle non-periodic boundary values effectively, making our Fourier methods even more versatile.
Application of Half-Range Expansions
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Now, let's dig into how these half-range expansions apply to real-world problems. Can someone remind me how we define half-range sine and cosine series?
Half-range sine series apply to odd functions with zero endpoints, and half-range cosine series apply to even functions with symmetry!
Exactly! In time-dependent problems, like the heat equation, sine series especially help model situations where we need to enforce boundary conditions of zero at the endpoints. Can anyone think of a physical context for this?
How about heat flow in a closed rod?
Perfect! In such cases, we model the temperature distribution using half-range sine expansions. This allows us to maintain the condition where the temperature at both ends of the rod is zero.
And what if we have a shape or function that is symmetric?
For symmetric situations, we utilize half-range cosine series. They fit scenarios like steady-state heat distribution in a two-dimensional plane.
To wrap up this session, remember, identifying whether youβre dealing with an odd or even function is crucial for applying the correct series.
Introduction & Overview
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Quick Overview
Standard
Half-range expansions utilize the Fourier series method for functions defined on intervals like 0 < x < L, focusing on odd or even extensions. This adaptation helps in fitting non-periodic boundary conditions into the Fourier framework, enhancing the applicability of Fourier series solutions in various physical scenarios.
Detailed
In this section on Half-Range Expansions, we explore how Fourier series, typically defined for periodic functions, can be adapted for functions that exist on a bounded interval. Specifically, when a function is defined between the bounds 0 < x < L, we can utilize half-range expansions to accommodate non-periodic boundary data. The use of half-range sine series is appropriate for odd extensions of functions (where the function is zero at the boundaries), while half-range cosine series are used for even extensions (where the function exhibits symmetry). This methodology allows for the effective application of Fourier series techniques to a broader variety of problems, particularly in partial differential equations related to heat, wave, and potential equations.
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Introduction to Half-Range Expansions
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If the function is defined on 0 < π₯ < πΏ instead of βπΏ < π₯ < πΏ, we use half-range expansions:
Detailed Explanation
Half-Range Expansions are techniques used in Fourier series when the function does not cover the full range of -L to L but is defined only from 0 to L. This situation typically arises in certain physical problems where the domain of interest is limited. The purpose of half-range expansions is to adapt functions that are only defined on a limited interval to fit the Fourier series format, which typically requires a periodic extension.
Examples & Analogies
Imagine a music piece that is composed to only exist in the notes D to G. If you wanted to play this music on a full piano (which represents the full range of -L to L), you would need to adapt those notes to cover the entire scale from C to B. In this case, half-range expansions allow you to consider only the 'essential' notes while creating an equivalent sound within a structured framework.
Half-Range Sine Series
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β’ Half-range sine series: for odd extensions (zero at ends).
Detailed Explanation
Half-range sine series are used when extending the function in an odd manner. This means that the function at the ends (0 and L) is set to zero. This type of extension is particularly useful when modeling situations like vibrations, where the ends of the string (or the medium concerned) are fixed and do not oscillate. Therefore, the sine series represents these kinds of physical environments accurately by reflecting the zeros at the boundaries.
Examples & Analogies
Think about a swing that is tied to a tree. At both endpoints, where the swing reaches its maximum height, it momentarily stops (like hitting zero). The swinging motion can be likened to a sine wave; at the edges (or boundaries), the motion is zero, just as the sine function equals zero at its peaks.
Half-Range Cosine Series
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β’ Half-range cosine series: for even extensions (symmetric boundary).
Detailed Explanation
Half-range cosine series are utilized for even extensions, which means the function is extended symmetrically about the boundary. In this scenario, the function behaves in such a way that it mirrors itself across the axis. This type of expansion is applicable in scenarios where the physical system is balanced or symmetric, such as heat distribution along a bar where ends are maintained at the same temperature.
Examples & Analogies
Consider a perfectly symmetrical seesaw balanced in the center. If you push down on one side, the other end reacts equally due to balance, creating a mirrored effect. This is similar to how the half-range cosine series mirrors the function at boundaries, preserving a symmetrical nature around a central point.
Purpose of Half-Range Expansions
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This allows fitting non-periodic boundary data into Fourier series format.
Detailed Explanation
The key purpose of using half-range expansions is to transform non-periodic boundary conditions into a form that can be managed with Fourier series. Non-periodic boundary data often arises in real-world problems where conditions or behaviors at the boundaries differ greatly. By fitting this data into the structure of Fourier series, we can apply the powerful methods of Fourier analysis to solve complex problems more effectively.
Examples & Analogies
Imagine trying to find the shape of a silhouette of a person standing against a wall. If the wall only allows you to see one half of their profile, you might still want to guess how the complete silhouette looks. Using half-range expansions is akin to completing that silhouette using the part you can see, thereby allowing for a full analysis even from limited information.
Definitions & Key Concepts
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Key Concepts
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Half-Range Expansion: Utilizing Fourier series for functions defined only on a limited interval.
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Half-Range Sine Series: Suitable for odd functions that return to zero at the endpoints.
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Half-Range Cosine Series: Suitable for even functions showing symmetry.
Examples & Real-Life Applications
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Examples
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Example of Half-Range Sine Series: Modeling temperature on a heated rod fixed at both ends.
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Example of Half-Range Cosine Series: Analyzing heat distribution in a circular pond.
Memory Aids
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π΅ Rhymes Time
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Sine is odd, and cosine's even; on limits set, their truth is leavened.
π Fascinating Stories
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Imagine a heat wave flowing through a rod; if the ends are cool, let sine be the nod. In a circle like a pond, cosine is strong; keep symmetry in mind, and youβll sing the right song.
π§ Other Memory Gems
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Remember 'S-O' for Sine and Odd, and 'C-E' for Cosine and Even - a simple aid to recall the types of expansions.
π― Super Acronyms
HRS
- Half-Range Sine for zero ends
- Half-Range Cosine for symmetrical friends.
Flash Cards
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Glossary of Terms
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Term: HalfRange Expansion
Definition:
An adaptation of Fourier series for functions defined on a limited interval, either by using sine or cosine series based on function symmetry.
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Term: HalfRange Sine Series
Definition:
A Fourier series for odd extensions of functions, used to represent a function defined on 0 < x < L with zero endpoints.
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Term: HalfRange Cosine Series
Definition:
A Fourier series for even extensions of functions, applied when the function exhibits symmetry.