Ed Series Lecture 6: Zero to L Interval

Introduction

• Channel: Pradiviri Academy
• Topic: Ed Series Lecture 6
• Interval: 0 to L
• Formula: There will be formulas similar to the previous ones

Formula Summary

• a0's formula: Instead of 1/2π use 1/L
• Cos term: cos(nπx/L)
• L-type identification: Compare with the question for the interval from 0 to 2L

Formula Detail

• a0: 1/2L ∫[0 to 2L] F(x) dx
• aN: 1/L ∫[0 to 2L] F(x)cos(nπx/L) dx
• bN: 1/L ∫[0 to 2L] F(x)sin(nπx/L) dx

Example

1. Question: Fourier expansion of a function

   • Interval: 0 to A
   • Technique: Compare 2L to given value to find L
   • A0 Calculation:
     • Substitute values, solve for A0
     • Example: a0 = A²/3

2. Finding An and Bn:

   • Use Integration by parts for complex functions
   • An Calculation:
     • Use the relevant f(x) value
     • Simplify and calculate
   • Bn Calculation:
     • Similar process, adjust for sin terms

3. Series Representation:

   • Use A0, An, Bn in Fourier series equation
   • Simplify to find the final series representation

Things to Note

• Correct Interval Identification: Compare with 0 to 2L or as given in the problem
• Minor Changes in Formula: Use L instead of π
• Cautions in Integration: Handle sign and cos/sin terms carefully
• Logic in Series: Substitute values correctly for the final series

Summary

• In this lecture, we understood the formulas of the Ed series and their intervals in detail.
• Learned to calculate various terms of the Fourier series through practical examples.
• Keep in mind that each function may have a different interval comparison, and it is necessary to adjust accordingly.