How do I find the Fourier Series for F(t) = sin(wt)?

In summary, the problem involves finding the Fourier series for the function F(t) = sin(wt) given a specific range of values for t. The solution involves using the Fourier series equation and evaluating the integral of sin(wt) using trigonometric identities. The solution process may be challenging, but with the help of trigonometric identities and manipulating the integral, a solution can be found.
  • #1
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Homework Statement



Consider F(t) = sin(wt) when 0 < t < pi/w and 0 when pi/w < t < 2pi/w. Where w is the frequency and t is the time. Find the Fourier Series

Homework Equations




F(t) = sum of (ck e^ikt)

See attached doc with math type; its a lot more readable.

The Attempt at a Solution




ck = (w/2pi integral from 0 to pi/w of sin(wt)*e^-ikx) + (w/2pi integral from pi/w to 2pi/w of sin(wt)*e^-ikx)

I’m not sure how to do this integration. If all the trig functions had the same argument, it would be do-able, but they don’t. w is a constant, right? And k isn’t, right? :confused: We never dealt with stuff like this back in calc. I’m very grateful for any help or suggestions. I tried to use trig addition formulas, but that didn’t work. Thanks! :smile:
 

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  • #2
Your integral is over t, not x, so it should have exp(-ikt).
Write sin(wt) as [exp(iwt)-exp(-iwt)]/2i, and do the exponential integrals.
 

FAQ: How do I find the Fourier Series for F(t) = sin(wt)?

What is the Fourier series?

The Fourier series is a mathematical function that represents a periodic signal as a sum of sine and cosine waves with different frequencies and amplitudes. It is a way to decompose a complex function into simpler, periodic components.

How is the Fourier series used in science?

The Fourier series is used in many scientific fields, such as physics, engineering, and mathematics. It is used to analyze and understand periodic phenomena, such as sound waves, electrical signals, and vibrations. It is also used in signal processing, image and sound compression, and solving differential equations.

How do you find the Fourier series of a function?

To find the Fourier series of a function, you need to determine its period, which is the length of one full cycle of the function. Then, you use integral calculus to find the coefficients of the sine and cosine terms in the series. These coefficients represent the amplitude and frequency of each component in the function.

What is the importance of the Fourier series?

The Fourier series is important because it allows us to represent complex, non-periodic functions as a sum of simpler, periodic functions. This simplifies the analysis of these functions and makes it easier to understand their behavior. It also has many practical applications in various fields of science and engineering.

Can all functions be represented by a Fourier series?

Not all functions can be represented by a Fourier series. The function must be periodic, meaning it repeats itself over a certain interval. If a function is not periodic, its Fourier series will not converge to the original function. However, there are other techniques, such as the Fourier transform, that can be used to analyze non-periodic functions.

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