Fourier Series of e^x from -pi to pi (fourier series engineering mathematics)

TL;DR

This video explains the process of finding the Fourier series representation of e^x on the interval -π to π, utilizing integration and trigonometric formulas.

Transcript

and in fact this right here is not periodic but we just want to consider that interval okay and then after all the work in fact in the end i have a really nice summation question for you guys so be sure you guys watch all the whole thing and if you guys like this video be sure to give me a like as well enough talking let's get started e to the x th... Read More

Key Insights

• ❓ The Fourier series for e^x on the interval -π to π involves calculating the coefficients a_n and b_n through integration.
• ❓ The convergence of the Fourier series at the endpoints of the interval should be analyzed separately.
• 👻 Plugging in x=0 allows for simplification and calculation of the Fourier series equation for e^x.
• 👨‍💼 The derived Fourier series equation represents e^x through a combination of coefficients, cosine functions, and sine functions.

Questions & Answers

Q: Does the Fourier series for e^x converge at the endpoints of the interval?

No, the Fourier series for e^x does not converge at the endpoints of the interval. Convergence at the endpoints should be analyzed separately.

Q: Can the Fourier series equation be simplified further?

The derived Fourier series equation for e^x on the interval -π to π cannot be simplified further. It includes terms with coefficients, cosine, and sine functions.

Q: How does plugging in x=0 help determine the convergence of the series?

Plugging in x=0 helps determine the convergence of the series at that specific value. In this case, substituting x=0 enables us to simplify the Fourier series equation for e^x and calculate its value.

Q: How can this Fourier series be used to find the sum of 1/(1+n^2)?

The Fourier series obtained in the video, involving the alternating version of -1^n/(1+n^2), can potentially be utilized to find the sum of 1/(1+n^2). Further analysis or manipulation of the series is necessary to achieve that result.

Summary & Key Takeaways

• The video demonstrates the derivation of the Fourier series representation of e^x on the interval -π to π.

• The process involves calculating the coefficients a_n and b_n through integrations of e^x over the interval.

• The final Fourier series for e^x on the interval is obtained by combining the derived coefficients with cosine and sine terms.