Chapter 1: Infinite Series, Power Series

As in Problem 1, solve

$\mathbf{2}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }\mathbf{x}\mathbf{\partial }\mathbf{y}}\mathbf{-}\mathbf{10}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }{\mathbf{y}}^{\mathbf{2}}}\mathbf{=}\mathbf{0}$

by making the change of variables $\mathit{u}\mathbf{=}\mathbf{5}\mathit{x}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{,}\mathit{v}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{+}\mathit{y}$.

Suppose that $\mathit{w}\mathbf{=}\mathit{f}\left(x,y\right)$satisfies

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{w}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{-}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{w}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\mathbf{1}$.

Put $\mathit{x}\mathbf{=}\mathit{u}\mathbf{+}\mathit{v}\mathbf{,}\mathit{y}\mathbf{=}\mathit{u}\mathbf{-}\mathit{v}$, and show that w satisfies $\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{w}}{\mathbf{\partial }\mathbf{u}\mathbf{\partial }\mathbf{v}}\mathbf{=}\mathbf{1}$. Hence solve the equation.

Verify the chain rule formulas

$\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{x}}\mathbf{=}\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{r}}\frac{\mathbf{\partial }\mathbf{r}}{\mathbf{\partial }\mathbf{x}}\mathbf{+}\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{\theta }}\frac{\mathbf{\partial }\mathbf{\theta }}{\mathbf{\partial }\mathbf{x}}$,

and similar formulas for

$\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{y}},\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{r}}\mathbf{,}\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{\theta }}$,

using differentials. For example, write

$\mathit{d}\mathit{F}\mathbf{=}\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{r}}\mathit{d}\mathit{r}\mathbf{+}\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{\theta }}\mathit{d}\mathit{\theta }$

and substitute for$\mathit{d}\mathit{r}$and $\mathit{d}\mathit{\theta }$:

$\mathit{d}\mathit{r}\mathbf{=}\frac{\mathbf{\partial }\mathbf{r}}{\mathbf{\partial }\mathbf{x}}\mathit{d}\mathit{x}\mathbf{+}\frac{\mathbf{\partial }\mathbf{r}}{\mathbf{\partial }\mathbf{y}}\mathit{d}\mathit{y}$(and similarly $\mathit{d}\mathit{\theta }$).

Collect coefficients of dx and dy; these are the values of $\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{x}}$ and $\frac{\mathbf{\partial }\mathbf{F}}{\mathbf{\partial }\mathbf{y}}$.

Solve equations (11.11) to get equations (11.12).

Reduce the equation

${\mathit{x}}^{\mathbf{2}}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{2}}}\mathbf{+}\mathbf{2}\mathit{x}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{-}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{0}$

to a differential equation with constant coefficients in $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{\mathbf{d}{\mathbf{z}}^{\mathbf{2}}}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{z}}$, and y by the change of variable $\mathit{x}\mathbf{=}{\mathit{e}}^{\mathbf{x}}$. (See Chapter 8, Section 7d.)

Find the interval of convergence, including end-point tests :$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}{\mathbf{x}}^{\mathbf{2}\mathbf{n}\mathbf{-}\mathbf{1}}}{\mathbf{2}\mathbf{n}\mathbf{-}\mathbf{1}}$

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

11. 0.678571428571428571…

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,$\mathbf{tan}\frac{\mathbf{1}}{\mathbf{z}}$

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.

$\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{{\mathbf{sin}}^{\mathbf{2}}\mathbf{2}\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}}$ .

Question: Show that the Maclaurin series for sin x converges to sin x . Hint: If f (x)= sin$f^{(n+1)}\left(X\right)=\pm \sin xor\pm \cos x$, and so $\left|f^{(n+1)}\left(x\right)\right|\le 1$

for all x and all n. Let$n\to \infty$in (14.2).