Chapter 12

Which of the following functions of $x$ could be represented by a Fourier series over the range indicated? (a) \(\tanh ^{-1}(x), \quad-\infty

Find the Fourier series of the function $f(x)=x$ in the range \(-\pi

Consider the function $f(x)=\exp (-x^2)$ in the range $0\le x\le 1.$ Show how it should be continued to give as its Fourier series a series (the actual form is not wanted) (a) with only cosine terms, (b) with only sine terms, (c) with period 1 and $(\mathrm{d})$ with period $2.$ Would there be any difference between the values of the last two series at (i) $x=0$, (ii) $x=1?$

Consider the representation as a Fourier series of the displacement of a string lying in the interval $0\le x\le L$ and fixed at its ends, when it is pulled aside by $y_0$ at the point $x=L/4$. Sketch the continuations for the region outside the interval that will (a) produce a series of period $L$, (b) produce a series that is antisymmetric about $x=0$, and (c) produce a series that will contain only cosine terms. (d) What are (i) the periods of the series in (b) and (c) and (ii) the value of the ${}^4{a}_0$-term' in (c)? (e) Show that a typical term of the series obtained in (b) is $$\frac{32y_0}{3n^2{\pi }^2}\sin \frac{n\pi }{4}\sin \frac{n\pi x}{L}$$

Show that the Fourier series for the function $y(x)=|x|$ in the range $-\pi \le x<\pi$ is $$y(x)=\frac{\pi }{2}-\frac{4}{\pi }\sum _{m=0}^{\mathrm{\infty }}\frac{\cos (2m+1)x}{(2m+1{)}^2}$$ By integrating this equation term by term from 0 to $x$, find the function $g(x)$ whose Fourier series is $$\frac{4}{\pi }\sum _{m=0}^{\mathrm{\infty }}\frac{\sin (2m+1)x}{(2m+1{)}^3}$$ Deduce the value of the sum $S$ of the series $$1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\cdots$$

By finding a cosine Fourier series of period 2 for the function $f(t)$ that takes the form $f(t)=\cosh (t-1)$ in the range $0\le t\le 1$, prove that $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2{\pi }^2+1}=\frac{1}{e^2-1}$$

Find the (real) Fourier series of period 2 for $f(x)=\cosh x$ and $g(x)=x^2$ in the range $-1\le x\le 1.$ By integrating the series for $f(x)$ twice, prove that $$\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n^2{\pi }^2(n^2{\pi }^2+1)}=\frac{1}{2}(\frac{1}{\sinh 1}-\frac{5}{6})$$

Show that the Fourier series for $|\sin \theta |$ in the range $-\pi \le \theta \le \pi$ is given by $$|\sin \theta |=\frac{2}{\pi }-\frac{4}{\pi }\sum _{m=1}^{\mathrm{\infty }}\frac{\cos 2m\theta }{4m^2-1}$$ By setting $\theta =0$ and $\theta =\pi /2$, deduce values for $$\sum _{m=1}^{\mathrm{\infty }}\frac{1}{4m^2-1}\text{ and }\sum _{m=1}^{\mathrm{\infty }}\frac{1}{16m^2-1}$$

Find the complex Fourier series for the periodic function of period $2\pi$ defined in the range $-\pi \le x\le \pi$ by $y(x)=\cosh x$. By setting $t=0$ prove that $$\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^2+1}=\frac{1}{2}(\frac{\pi }{\sinh \pi }-1)$$

The repeating output from an electronic oscillator takes the form of a sine wave $f(t)=\sin t$ for $0\le t\le \pi /2;$ it then drops instantaneously to zero and starts again. The output is to be represented by a complex Fourier series of the form $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{4nti}$$ Sketch the function and find an expression for $c_n$. Verify that $c_{-n}=c_n^{\ast }.$ Demonstrate that setting $t=0$ and $t=\pi /2$ produces differing values for the sum $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{16n^2-1}$$ Determine the correct value and check it using the quoted result of exercise $12.5.$