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Chapter 10

# MATH-M 344 Chapter 10: M344 10.3 Notes (Mar. 24) Premium

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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0 + βπ=1 ππcos( ππ π₯) + ππsin( ππ π₯) converge for each value of π₯? 2 πΏ πΏ ο· Recall piecewise continuous function- there exists a partition on π,π π 0 π₯ <1π₯ < β― < π₯π= π so that, for function π defined on π,π : o π continuous on π₯ ππ₯ π+1)for π = 0,β¦,π β 1 o π approaches finite limit at endpoints of each subintervπl π+1π₯ ο§ lim π(π₯) and lim π(π₯) are finite, where π₯ π π₯ ,π₯ ) π₯βπ₯π π₯βπ₯ π+1 π π+1 o Ex. Piecewise continuous o Ex. Not piecewise continuous π₯,0 β€ π₯ < 1 o Ex. π π₯ = { 2 is piecewise continuous (π₯ β 1 ,1 β€ π₯ β€ 2 1,β1 β€ π₯ < 0 π₯ o Ex. π π₯ = { 0,π₯ = 0 not piecewise continuous 1 π₯,0 < π₯ β€ 1 ο· Notation (if π π π,π ): ( ) o π π + = liπ₯βπ+π₯) ο§ π₯ decreases toward π from right ο§ Corresponds to π₯βπ₯π+1π₯) = π(π₯ π+1β) o π π β = liπ₯βπβπ₯ ( ) ο§ π₯ increases toward π from left ο§ Corresponds to π₯βπ₯ π(π₯) = π(π₯ π) π ο· Ex. π π₯ = { π₯,0 β€ π₯ < 1 (π₯ β 1 ,1 β€ π₯ β€ 2 o π 1 + = lim π₯ β 1 )2= 0 π₯β1+ o π 1 β = lim π₯ = 1 π₯β1β ο· Theorem 10.3.1- Fourier Convergence Theorem o Suppose function π is 2πΏ-periodic and that π and πβ² are piecewise continuous π ππ ππ o Then, at all π₯ where π continuous, π π₯ + β π=1 ππcos( π₯) + ππsin( π₯), where 1 πΏ 2 1 πΏ ππ πΏ Fourier coefficien0s π β« π(π₯)ππ₯, π π = β« π π₯ cos( π₯)ππ₯, and ππ= 1 πΏ ππ πΏ βπΏ πΏ βπΏ πΏ β« π π₯ sin( π₯)ππ₯ πΏ βπΏ πΏ ( ) ( ) o For some π₯ where π discontinuous,+ β π=1π πos( ππ π₯) + ππsin( πππ₯) = π π₯β +π π₯+ = 2 πΏ πΏ 2 mean of π(π₯β) and π(π₯+) π π₯β +π π₯+ ο§ If π discontinuous at π₯, then Fourier series converge2 to π π₯β +π π₯+) o Can conclude that iff π continuous at π₯2 = π(π₯) β1,β1 β€ π₯ < 0 ο· Ex. π π₯ = { , π piece
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