Chapter 1: Infinite Series, Power Series

Show that if $\mathit{C}$is a matrix whose columns are the components $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$of two perpendicular vectors each of unit length, then $\mathit{C}$is an orthogonal matrix. Hint: Find${\mathit{C}}^{\mathbf{T}}\mathit{C}$

Test the following series for convergence

$\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}\mathbf{n}}{\mathbf{n}\mathbf{+}\mathbf{5}}$

Use equation (1.8) to find the fraction that are equivalent to following repeating decimals.

6. 0.61111...

Find out whether infinity is a regular point, an essential singularity, or a pole (and if a pole, of what order) for each of the following functions. Find the residue of each function at infinity $\frac{{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{5}}{\mathbf{2}}$

$\mathit{x}\sqrt{\mathbf{1}\mathbf{+}\mathbf{x}}$

To find the maximum and the minimum points of the given function.

x³-y³- 2xy+2

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{{\mathbf{d}}^{\mathbf{3}}}{{\mathbf{dx}}^{\mathbf{3}}}\left(\frac{x^2{e}^x}{1-x}\right)$at x=0 .

(a) Find the Fourier series of period 2$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\left(x-1\right)}^{\mathbf{2}}$on (0,2)$$" width="9" height="19" role="math">$$" width="9" height="19" role="math">" width="9" height="19" role="math">

(b) Use your result in (a) to evaluate$\mathbf{\sum }_{}^{}\mathbf{1}\mathbf{/}{\mathit{n}}^{\mathbf{4}}$.

Given

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\left\{\begin{array}{c}1,\\ -1,\end{array}\begin{array}{ccc}-2& <x& <0\\ 0& <x& <2\end{array}\right\}$

find the exponential Fourier transform $\mathit{g}\left(\alpha \right)$and the sine transform${\mathit{g}}_{\mathbf{s}}\left(\alpha \right)$ .Write f(x) as an integral and use your result to evaluate

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{\left(cos2\alpha -1\right)\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{2}\mathbf{\alpha }}{\mathbf{\alpha }}\mathit{d}\mathit{\alpha }$

Given

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\left(\begin{array}{c}x,\\ 2-x,\\ 0\end{array}\begin{array}{ccccc}0& \le & x& \le & 1\\ 1& \le & x& \le & 2\\ & x& \ge & 2& \end{array}\right)$

find the cosine transform of f(x) and f(x) use it to write as an integral. Use your result to evaluate

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{2}}\mathbf{\alpha }\mathbf{}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\alpha }\mathbf{/}\mathbf{2}}{{\mathbf{\alpha }}^{\mathbf{2}}}\mathit{d}\mathit{\alpha }$