Homework Seven

65

$$\lambda>0, y_n=\sin((n+\frac{1}{2})\pi x), \lambda_n = (n+\frac{1}{2})^2.$$

$$\lambda=0, y\equiv 0, {\rm no} {\rm eigenfunction},$$

$$\lambda<0, y\equiv 0, {\rm no} {\rm eigenfunction}.$$

66

$$\lambda>0, y_n=\cos(n \pi x/L), \lambda_n = (n \pi/L)^2.$$

$$\lambda=0, y=1,$$

$$\lambda<0, y\equiv 0, {\rm no} {\rm eigenfunction}.$$

67

$$f(x) = \frac{1}{2} - \frac{2}{\pi}\sum\limits_{m=0}^\infty \frac{\sin(\frac{ (2 m+1)\pi x}{L})}{2 m +1}.$$

68

$$f(x) = \frac{2}{\pi}\sum\limits_{n=1}^\infty \frac{ (-1)^{n+1}}{n}\sin(n\pi x).$$

69

$$f(x) = \frac{1}{2}- \frac{1}{\pi}\sum\limits_{m=1}^\infty\frac{\sin(2m\pi x)}{m}.$$

70

$$f(x) = \frac{1}{2}+ \frac{4}{\pi^2}\sum\limits_{n=0}^\infty\frac{\cos((2n+1)\pi x)}{2 n+1}.$$

71

$$f(x) = \frac{3 l}{4}+ \sum\limits_{n=1}^\infty\Big[ \frac{ 2... ...^2}+ \frac{ (-1)^{n+1} l \sin[\frac{ n \pi x}{l}]}{ n\pi}\Big].$$

72

$$f(x) = -x, -l<x<l,$$

$$f(x+2 l ) = f(x),$$

$$f(x) = 2 l - x, l < x< 2 l,$$

$$f(x) = -2 l - x, - 3 l < x< - 2 l.$$

73

$$f(x) = s + \frac{2}{\pi}\sum_{n=0}^\infty \frac{\sin n \pi s \cos n \pi x}{n}.$$

74

$$f(x) = \frac{2}{3} - \frac{4}{\pi^2}\sum\limits_{n=1}^\infty (-1)^n \frac{\cos n \pi x}{n^2}.$$

75

$$a_0 = \frac{1}{3},$$

$$a_n = \frac{2}{ (n\pi)^2}(-1)^n,$$

$$b_n = -\frac{1}{n\pi}, {\rm n is even},$$

$$b_n = \frac{1}{n\pi} - \frac{4}{(n\pi)^3}, {\rm n is odd.}$$

76

Even extension:

$$f(x) = \frac{1}{4} + \frac{4}{\pi^2}\sum\limits_{n=1}^\infty \frac{1-\cos\frac{n \pi}{2}}{n^2}\cos\frac{n \pi x}{2}.$$

Odd extension:

$$f(x)=\frac{4}{\pi^2}\sum\limits_{n=1}^{\infty} \frac{\frac{n\pi}{2}-\sin\frac{n\pi}{2}}{n^2}\sin(\frac{n\pi x}{2}).$$

77

$$f(x) = \sum\limits_{n=0}^\infty\frac{2}{n\pi}\Big(-\cos(n\pi)+\frac{2}{n\pi} \sin\frac{n\pi}{2}\Big)\sin\frac{n\pi x}{2}.$$

78

$$f(x)=1, 0\le x<\pi,$$

Cosine Series, period $2\pi$,

$$a_0 = 2, a_n=0, n = 1,2,\dots .$$

79

$$f(x)=1, 0\le x<\pi,$$

Sine Series, period $2\pi$,

$$f(x) = \frac{4}{\pi}\sum\limits_{m = 0}^\infty \frac{\sin (2 m+1)x }{2m+1}.$$

80

$$f(x) = l - x, 0\le x\le l,$$

Cosine series, period $2l$,

$$f(x) = \frac{l}{2} + \frac{4 l}{\pi^2}\sum\limits_{m = 0}^\infty \frac{\cos \frac{(2 m+1)\pi x }{l}}{(2 m+1)^2}.$$

81

$$f(x) = l - x, 0<x<l,$$

Sine Series, Period $2 l,$

$$f(x)=\frac{2 l }{\pi}\sum\limits_{n=1}^{\infty}\frac{\sin\frac{n\pi x}{l}}{n}.$$