Proof of the Orthogonality of the Set of Trigonometric Functions
📂Fourier AnalysisProof of the Orthogonality of the Set of Trigonometric Functions
Theorem
The set {1, cosLπx, cosL2πx,⋯, sinLπx, sinL2πx, ⋯} of functions 2L that are periodic functions is an orthogonal set in the interval [−L, L). In other words, for m,n=1,2,3,…, the following holds.
L1∫−LLcosLmπxcosLnπxdxL1∫−LLsinLmπxsinLnπxdx∫−LLcosLmπxsinLnπxdxL1∫−LLcosLnπxdxL1∫−LLsinLnπxdx=δmn=δmn=0=0=0
Here, δ is the Kronecker delta.
Corollary
According to (4),(5), the average of one period of cosine and sine is 0.
Explanation
Due to the Euler’s formula, the set of exponential functions also possesses orthogonality. This fact is significant in Fourier analysis as it enables the representation of periodic functions as series of periodic functions, namely the Fourier series.
Proof
(1)
∫−LLcosLmπxcos Lnπxdx(1)
∫−LLcosLmπxcosLnπxdx(m,n=1,2,…m=n)=21∫−LL[cosL(m+n)πx+cosL(m−n)πx]dx=21[(m+n)πLsinL(m+n)πx]−LL+21[(m−n)πLsinL(m−n)πx]−LL=21[(m+n)πLsin((m+n)π)+(m+n)πLsin((m+n)π)]+21[(m−n)πLsin((m−n)π)+(m−n)πLsin((m−n)π)]=0
The first equality holds due to the product-to-sum identities of trigonometric functions. The last equality holds because m+n, m−n is an integer not equal to 0, so all terms are 0.
case 1.2 m=n
∫−LL(cosLmπx)2dx=21∫−LL(1+cosL2mπx)dx=21[x+2mπLsinL2mπx]−LL=21(2L)=L⟹L1∫−LL(cosLmπx)2dx=1
The first equality holds due to the half-angle identities of trigonometric functions.
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(2)
∫−LLsinLmπxsinLnπxdx(2)
case 2.1 m=n
∫−LLsinLmπxsinLnπxdx (m,n=1,2,⋯,m=n)=21∫−LL[cosL(m−n)πx−cosL(m+n)πx]dx=21[(m−n)πLsinL(m−n)πx]−LL−21[(m+n)πLsinL(m+n)πx]−LL=21[(m−n)πLsin((m−n)π)+(m−n)πLsin((m+n)π)]−21[(m+n)πLsin((m+n)π)+(m+n)πLsin((m+n)π)]=0
The first equality holds due to the product-to-sum identities of trigonometric functions. The last equality holds for the same reason as in case 1.1, since all terms are 0.
case 2.2 m=n
∫−LL(sinLmπx)2dx=21∫−LL(1−cosL2mπx)dx=21[x−2mπLsinL2mπx]−LL=21(2L)=L⟹L1∫−LL(sinLmπx)2dx=1
The first equality holds due to the half-angle identities of trigonometric functions.
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(3)
∫−LLcosLmπxsinLnπxdx(3)
case 3.1 m=n
∫−LLcosLmπxsinLnπxdx=21∫−LL[sinL(m+n)πx−sinL(m−n)πx]dx=21[−(m+n)πLcosL(m+n)πx]−LL−21[−(m−n)πLcosL(m−n)πx]−LL=21[−(m+n)πLcos((m+n)π)+(m+n)πLcos((m+n)π)]−21[−(m−n)πLcos((m−n)π)+(m−n)πLcos((m−n)π)]=0
The first equality holds due to the product-to-sum identities of trigonometric functions.
case 3.2 m=n
∫−LLcosLmπxsinLnπxdx=∫−LLcosLmπxsinLmπxdx=21∫−LL2cosLmπxsinLmπxdx=21∫−LLsinL2mπxdx=21(−2mπLcos2mπ+2mπLcos2mπ)=0
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(4),(5)
∫−LLcosLnπxdx=[nπLsinLnπx]−LL=nπLsinnπ+nπLsinnπ=0
The last equality holds because n is an integer. The sine function also follows for the same reason
∫−LLsinLnπxdx=[−nπLcosLnπx]−LL=−nπLcosnπ+nπLcosnπ=0
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