직교성(Orthogonality) #1

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$$\langle f,g\rangle=\int_{a}^{b}{r(x)f(x)g(x)dx}=0$$

Here, \(r(x)~\)is nonnegative weight function and we usually put them 1.

$$\langle f,g \rangle=\int_{a}^{b}{f(x)g(x)dx}=0$$

 

Simple example of orthogonality 1:

$$\int_{0}^{2\pi}{\sin x\cos xdx}=0$$

 

$$u=\sin x$$

$$\frac{du}{dx}=\cos x$$

$$du=\cos x dx$$

$$\sin(x=0)=0~~~~\sin(x=2\pi)=0$$

$$\int_{0}^{0}{u du}=0$$

 

Simple example of orthogonality 2:

$$\int_{0}^{2\pi}{\sin(mx)\sin(nx)dx}=0$$

$$\sin(a)\sin(b)=\frac{1}{2}\left\{\cos(a-b)-\cos(a+b)\right\}$$

$$\int_{0}^{2\pi}{\sin(mx)\sin(nx)dx}=\frac{1}{2}\int_{0}^{2\pi}{\left[\cos(mx-nx)-\cos(mx+nx)\right]dx}=0$$

$$=\frac{1}{2}\int_{0}^{2\pi}{\cos(mx-nx)dx}-\frac{1}{2}\int_{0}^{2\pi}{\cos(mx+nx)dx}$$

 

$$u=(m-n)x$$

$$\frac{du}{dx}=m-n$$

$$dx=\frac{du}{m-n}$$

 

$$\int_{0}^{2\pi}{\cos(mx-nx)dx}=\int_{0}^{2\pi}{\cos(u)\frac{du}{m-n}}=\frac{1}{m-n}\int_{0}^{2\pi}{\cos(u)du}$$

$$=-\frac{1}{m-n}\sin(u)+c_{1}$$

$$=-\frac{1}{m-n}\sin(m-n)x+c_{1}$$

 

$$v=(m+n)x$$

$$\frac{dv}{dx}=m+n$$

$$dx=\frac{dv}{m+n}$$

 

$$\int_{0}^{2\pi}{\cos(mx+nx)dx}=\int_{0}^{2\pi}{\cos(v)\frac{dv}{m+n}}=\frac{1}{m+n}\int_{0}^{2\pi}{\cos(v)dv}$$

$$=-\frac{1}{m+n}\sin(v)+c_{2}$$

$$=-\frac{1}{m+n}\sin(m+n)x+c_{2}$$

 

$$-\frac{1}{2}\left[\frac{1}{m+n}\sin(m+n)x+\frac{1}{m-n}\sin(m-n)x\right]_{0}^{2\pi}+c_{3}=0~~(m\neq n)$$

 

애초에 \(sin\) 그래프가 해당 구간 내에서는 0이며 적분상수인 \(c_{3}\)는 무시해도 무방하다.

 

Wikipedia, Orthogonality

Wikipedia, Orthogonal functions

Wikipedia, Orthogonal polynomials

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