MATH-M 344 Chapter 10: M344 10.3 Notes (Mar. 24)
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Department
Mathematics
Course
MATH-M 344
Professor
Benjamin Melinand
Semester
Spring

Description
M344 Section 10.3 Notes- The Fourier Series Theorem 3-24-17 ο‚· When does the Fourier seriesπ‘Ž0 + βˆ‘π‘š=1 π‘Žπ‘šcos( π‘šπœ‹ π‘₯) + π‘π‘šsin( π‘šπœ‹ π‘₯) converge for each value of π‘₯? 2 𝐿 𝐿 ο‚· Recall piecewise continuous function- there exists a partition on π‘Ž,𝑏 π‘Ž 0 π‘₯ <1π‘₯ < β‹― < π‘₯𝑛= 𝑏 so that, for function 𝑓 defined on π‘Ž,𝑏 : o 𝑓 continuous on π‘₯ 𝑖π‘₯ 𝑖+1)for 𝑖 = 0,…,𝑛 βˆ’ 1 o 𝑓 approaches finite limit at endpoints of each subinterv𝑖l 𝑖+1π‘₯  lim 𝑓(π‘₯) and lim 𝑓(π‘₯) are finite, where π‘₯ πœ– π‘₯ ,π‘₯ ) π‘₯β†’π‘₯𝑖 π‘₯β†’π‘₯ 𝑖+1 𝑖 𝑖+1 o Ex. Piecewise continuous o Ex. Not piecewise continuous π‘₯,0 ≀ π‘₯ < 1 o Ex. 𝑓 π‘₯ = { 2 is piecewise continuous (π‘₯ βˆ’ 1 ,1 ≀ π‘₯ ≀ 2 1,βˆ’1 ≀ π‘₯ < 0 π‘₯ o Ex. 𝑓 π‘₯ = { 0,π‘₯ = 0 not piecewise continuous 1 π‘₯,0 < π‘₯ ≀ 1 ο‚· Notation (if 𝑐 πœ– π‘Ž,𝑏 ): ( ) o 𝑓 𝑐 + = liπ‘₯→𝑐+π‘₯)  π‘₯ decreases toward 𝑐 from right  Corresponds to π‘₯β†’π‘₯𝑖+1π‘₯) = 𝑓(π‘₯ 𝑖+1βˆ’) o 𝑓 𝑐 βˆ’ = liπ‘₯β†’π‘βˆ’π‘₯ ( )  π‘₯ increases toward 𝑐 from left  Corresponds to π‘₯β†’π‘₯ 𝑓(π‘₯) = 𝑓(π‘₯ 𝑖) 𝑖 ο‚· Ex. 𝑓 π‘₯ = { π‘₯,0 ≀ π‘₯ < 1 (π‘₯ βˆ’ 1 ,1 ≀ π‘₯ ≀ 2 o 𝑓 1 + = lim π‘₯ βˆ’ 1 )2= 0 π‘₯β†’1+ o 𝑓 1 βˆ’ = lim π‘₯ = 1 π‘₯β†’1βˆ’ ο‚· Theorem 10.3.1- Fourier Convergence Theorem o Suppose function 𝑓 is 2𝐿-periodic and that 𝑓 and 𝑓′ are piecewise continuous π‘Ž π‘šπœ‹ π‘šπœ‹ o Then, at all π‘₯ where 𝑓 continuous, 𝑓 π‘₯ + βˆ‘ π‘š=1 π‘Žπ‘šcos( π‘₯) + π‘π‘šsin( π‘₯), where 1 𝐿 2 1 𝐿 π‘›πœ‹ 𝐿 Fourier coefficien0s π‘Ž ∫ 𝑓(π‘₯)𝑑π‘₯, π‘Ž π‘š = ∫ 𝑓 π‘₯ cos( π‘₯)𝑑π‘₯, and π‘π‘š= 1 𝐿 π‘›πœ‹ 𝐿 βˆ’πΏ 𝐿 βˆ’πΏ 𝐿 ∫ 𝑓 π‘₯ sin( π‘₯)𝑑π‘₯ 𝐿 βˆ’πΏ 𝐿 ( ) ( ) o For some π‘₯ where 𝑓 discontinuous,+ βˆ‘ π‘š=1π‘Ž π‘šos( π‘šπœ‹ π‘₯) + π‘π‘šsin( π‘šπœ‹π‘₯) = 𝑓 π‘₯βˆ’ +𝑓 π‘₯+ = 2 𝐿 𝐿 2 mean of 𝑓(π‘₯βˆ’) and 𝑓(π‘₯+) 𝑓 π‘₯βˆ’ +𝑓 π‘₯+  If 𝑓 discontinuous at π‘₯, then Fourier series converge2 to 𝑓 π‘₯βˆ’ +𝑓 π‘₯+) o Can conclude that iff 𝑓 continuous at π‘₯2 = 𝑓(π‘₯) βˆ’1,βˆ’1 ≀ π‘₯ < 0 ο‚· Ex. 𝑓 π‘₯ = { , 𝑓 piece
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