# 5.7: Problems

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## Exercise $$\PageIndex{1}$$

Consider the set of vectors $$(-1,1,1),(1,-1,1),(1,1,-1)$$.

1. Use the Gram-Schmidt process to find an orthonormal basis for $$R^{3}$$ using this set in the given order.
2. What do you get if you do reverse the order of these vectors?

## Exercise $$\PageIndex{2}$$

Use the Gram-Schmidt process to find the first four orthogonal polynomials satisfying the following:

1. Interval: $$(-\infty, \infty)$$ Weight Function: $$e^{-x^{2}}$$.
2. Interval: $$(0, \infty)$$ Weight Function: $$e^{-x}$$.

## Exercise $$\PageIndex{3}$$

Find $$P_{4}(x)$$ using

1. The Rodrigues’ Formula in Equation (5.3.3).
2. The three term recursion formula in Equation (5.3.5).

## Exercise $$\PageIndex{4}$$

In Equations (5.3.18)-(5.3.25) we provide several identities for Legendre polynomials. Derive the results in Equations (5.3.19)-(5.3.25) as described in the text. Namely,

1. Differentiating Equation (5.3.18) with respect to $$x$$, derive Equation (5.3.19).
2. Derive Equation (5.3.20) by differentiating $$g(x, t)$$ with respect to $$x$$ and rearranging the resulting infinite series.
3. Combining the last result with Equation (5.3.18), derive Equations (5.3.21)-(5.3.22).
4. Adding and subtracting Equations (5.3.21)-(5.3.22), obtain Equations (5.3.23)-(5.3.24).
5. Derive Equation (5.3.25) using some of the other identities.

## Exercise $$\PageIndex{5}$$

Use the recursion relation (5.3.5) to evaluate $$\int_{-1}^{1} x P_{n}(x) P_{m}(x) d x, n \leq m$$.

## Exercise $$\PageIndex{6}$$

Expand the following in a Fourier-Legendre series for $$x \in(-1,1)$$.

1. $$f(x)=x^{2}$$.
2. $$f(x)=5 x^{4}+2 x^{3}-x+3$$.
3. $$f(x)=\left\{\begin{array}{cc}-1, & -1<x<0, \\ 1, & 0<x<1 .\end{array}\right.$$
4. $$f(x)=\left\{\begin{array}{cc}x, & -1<x<0, \\ 0, & 0<x<1 .\end{array}\right.$$

## Exercise $$\PageIndex{7}$$

Use integration by parts to show $$\Gamma(x+1)=x \Gamma(x)$$.

## Exercise $$\PageIndex{8}$$

Prove the double factorial identities: $(2 n) ! !=2^{n} n !\nonumber$ and $(2 n-1) ! !=\frac{(2 n) !}{2^{n} n !} .\nonumber$

## Exercise $$\PageIndex{9}$$

Express the following as Gamma functions. Namely, noting the form $$\Gamma(x+1)=\int_{0}^{\infty} t^{x} e^{-t} d t$$ and using an appropriate substitution, each expression can be written in terms of a Gamma function.

1. $$\int_{0}^{\infty} x^{2 / 3} e^{-x} d x$$.
2. $$\int_{0}^{\infty} x^{5} e^{-x^{2}} d x$$
3. $$\int_{0}^{1}\left[\ln \left(\frac{1}{x}\right)\right]^{n} d x$$

## Exercise $$\PageIndex{10}$$

The coefficients $$C_{k}^{p}$$ in the binomial expansion for $$(1+x)^{p}$$ are given by $C_{k}^{p}=\frac{p(p-1) \cdots(p-k+1)}{k !} .\nonumber$

1. Write $$C_{k}^{p}$$ in terms of Gamma functions.
2. For $$p=1 / 2$$ use the properties of Gamma functions to write $$C_{k}^{1 / 2}$$ in terms of factorials.
3. Confirm you answer in part $$b$$ by deriving the Maclaurin series expansion of $$(1+x)^{1 / 2}$$.

## Exercise $$\PageIndex{11}$$

The Hermite polynomials, $$H_{n}(x)$$, satisfy the following:

1. $$\left\langle H_{n}, H_{m}\right\rangle=\int_{-\infty}^{\infty} e^{-x^{2}} H_{n}(x) H_{m}(x) d x=\sqrt{\pi} 2^{n} n ! \delta_{n, m}$$.
2. $$H_{n}^{\prime}(x)=2 n H_{n-1}(x)$$.
3. $$H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x)$$.
4. $$H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}}\left(e^{-x^{2}}\right)$$.

Using these, show that

1. $$H_{n}^{\prime \prime}-2 x H_{n}^{\prime}+2 n H_{n}=0$$. [Use properties ii. and iii.]
2. $$\int_{-\infty}^{\infty} x e^{-x^{2}} H_{n}(x) H_{m}(x) d x=\sqrt{\pi} 2^{n-1} n !\left[\delta_{m, n-1}+2(n+1) \delta_{m, n+1}\right]$$. [Use properties i. and iii.]
3. $$H_{n}(0)=\left\{\begin{array}{cc}0, & n \text { odd, } \\ (-1)^{m} \frac{(2 m) !}{m !}, & n=2 m .\end{array} \quad\right.$$ [Let $$x=0$$ in iii. and iterate. Note from iv. that $$H_{0}(x)=1$$ and $$H_{1}(x)=2 x$$. ]

## Exercise $$\PageIndex{12}$$

In Maple one can type simplify(LegendreP $$\left(2^{*} \mathrm{n}-2,0\right)$$-LegendreP $$\left(2^{*} \mathrm{n}, 0\right)$$ ); to find a value for $$P_{2 n-2}(0)-P_{2 n}(0)$$. It gives the result in terms of Gamma functions. However, in Example 5.3.8 for Fourier-Legendre series, the value is given in terms of double factorials! So, we have $P_{2 n-2}(0)-P_{2 n}(0)=\frac{\sqrt{\pi}(4 n-1)}{2 \Gamma(n+1) \Gamma\left(\frac{3}{2}-n\right)}=(-1)^{n} \frac{(2 n-3) ! !}{(2 n-2) ! !} \frac{4 n-1}{2 n} .\nonumber$ You will verify that both results are the same by doing the following:

1. Prove that $$P_{2 n}(0)=(-1)^{n} \frac{(2 n-1) ! !}{(2 n) ! !}$$ using the generating function and a binomial expansion.
2. Prove that $$\Gamma\left(n+\frac{1}{2}\right)=\frac{(2 n-1) ! !}{2^{n}} \sqrt{\pi}$$ using $$\Gamma(x)=(x-1) \Gamma(x-1)$$ and iteration.
3. Verify the result from Maple that $$P_{2 n-2}(0)-P_{2 n}(0)=\frac{\sqrt{\pi}(4 n-1)}{2 \Gamma(n+1) \Gamma\left(\frac{3}{2}-n\right)}$$.
4. Can either expression for $$P_{2 n-2}(0)-P_{2 n}(0)$$ be simplified further?

## Exercise $$\PageIndex{13}$$

A solution Bessel’s equation, $$x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-n^{2}\right) y=0,$$, can be found using the guess $$y(x)=\sum_{j=0}^{\infty} a_{j} x^{j+n}$$. One obtains the recurrence relation $$a_{j}=\frac{-1}{(2 n+i)} a_{j-2}$$. Show that for $$a_{0}=\left(n ! 2^{n}\right)^{-1}$$ we get the Bessel function of the first kind of order $$n$$ from the even values $$j=2 k$$ :$J_{n}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !(n+k) !}\left(\frac{x}{2}\right)^{n+2 k} .\nonumber$

## Exercise $$\PageIndex{14}$$

Use the infinite series in the last problem to derive the derivative identities (5.5.15) and (5.5.5):

1. $$\frac{d}{d x}\left[x^{n} J_{n}(x)\right]=x^{n} J_{n-1}(x) .$$
2. $$\frac{d}{d x}\left[x^{-n} J_{n}(x)\right]=-x^{-n} J_{n+1}(x) .$$

## Exercise $$\PageIndex{15}$$

Prove the following identities based on those in the last problem.

1. $$J_{p-1}(x)+J_{p+1}(x)=\frac{2 p}{x} J_{p}(x)$$.
2. $$J_{p-1}(x)-J_{p+1}(x)=2 J_{p}^{\prime}(x)$$.

## Exercise $$\PageIndex{16}$$

Use the derivative identities of Bessel functions, (5.5.15)-(5.5.5), and integration by parts to show that $\int x^{3} J_{0}(x) d x=x^{3} J_{1}(x)-2 x^{2} J_{2}(x)+C .\nonumber$

## Exercise $$\PageIndex{17}$$

Use the generating function to find $$J_{n}(0)$$ and $$J_{n}^{\prime}(0)$$.

## Exercise $$\PageIndex{18}$$

Bessel functions $$J_{p}(\lambda x)$$ are solutions of $$x^{2} y^{\prime \prime}+x y^{\prime}+\left(\lambda^{2} x^{2}-p^{2}\right) y=0$$. Assume that $$x \in(0,1)$$ and that $$J_{p}(\lambda)=0$$ and $$J_{p}(0)$$ is finite.

1. Show that this equation can be written in the form $\frac{d}{d x}\left(x \frac{d y}{d x}\right)+\left(\lambda^{2} x-\frac{p^{2}}{x}\right) y=0 \text {. }\nonumber$ This is the standard Sturm-Liouville form for Bessel’s equation.
2. Prove that $\int_{0}^{1} x J_{p}(\lambda x) J_{p}(\mu x) d x=0, \quad \lambda \neq \mu\nonumber$ by considering $\int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] d x .\nonumber$ Thus, the solutions corresponding to different eigenvalues $$(\lambda, \mu)$$ are orthogonal.
3. Prove that $\int_{0}^{1} x\left[J_{p}(\lambda x)\right]^{2} d x=\frac{1}{2} J_{p+1}^{2}(\lambda)=\frac{1}{2} J_{p}^{\prime 2}(\lambda) .\nonumber$

## Exercise $$\PageIndex{19}$$

We can rewrite Bessel functions, $$J_{v}(x)$$, in a form which will allow the order to be non-integer by using the gamma function. You will need the results from Problem $$\PageIndex{12}$$b for $$\Gamma\left(k+\frac{1}{2}\right)$$.

1. Extend the series definition of the Bessel function of the first kind of order $$v, J_{v}(x)$$, for $$v \geq 0$$ by writing the series solution for $$y(x)$$ in Problem $$\PageIndex{13}$$ using the gamma function.
2. Extend the series to $$J_{-v}(x)$$, for $$v \geq 0$$. Discuss the resulting series and what happens when $$v$$ is a positive integer.
3. Use these results to obtain the closed form expressions \begin{aligned} &J_{1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \sin x, \\ &J_{-1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \cos x . \end{aligned}
4. Use the results in part $$c$$ with the recursion formula for Bessel functions to obtain a closed form for $$J_{3 / 2}(x)$$.

## Exercise $$\PageIndex{20}$$

In this problem you will derive the expansion $x^{2}=\frac{c^{2}}{2}+4 \sum_{j=2}^{\infty} \frac{J_{0}\left(\alpha_{j} x\right)}{\alpha_{j}^{2} J_{0}\left(\alpha_{j} c\right)}, \quad 0<x<c,\nonumber$ where the $$\alpha_{j}^{\prime} s$$ are the positive roots of $$J_{1}(\alpha c)=0$$, by following the below steps.

1. List the first five values of $$\alpha$$ for $$J_{1}(\alpha c)=0$$ using the Table 5.5.1 and Figure 5.5.1. [Note: Be careful determining $$\alpha_{1}$$.]
2. Show that $$\left\|J_{0}\left(\alpha_{1} x\right)\right\|^{2}=\frac{c^{2}}{2}$$. Recall, $\left\|J_{0}\left(\alpha_{j} x\right)\right\|^{2}=\int_{0}^{c} x J_{0}^{2}\left(\alpha_{j} x\right) d x .\nonumber$
3. Show that $$\left\|J_{0}\left(\alpha_{j} x\right)\right\|^{2}=\frac{c^{2}}{2}\left[J_{0}\left(\alpha_{j} c\right)\right]^{2}, j=2,3, \ldots$$. (This is the most involved step.) First note from Problem $$\PageIndex{18}$$ that $$y(x)=J_{0}\left(\alpha_{j} x\right)$$ is a solution of $x^{2} y^{\prime \prime}+x y^{\prime}+\alpha_{j}^{2} x^{2} y=0 .\nonumber$
1. Verify the Sturm-Liouville form of this differential equation: $$\left(x y^{\prime}\right)^{\prime}=-\alpha_{j}^{2} x y .$$
2. Multiply the equation in part i. by $$y(x)$$ and integrate from $$x=0$$ to $$x=c$$ to obtain \begin{align} \int_{0}^{c}\left(x y^{\prime}\right)^{\prime} y d x &=-\alpha_{j}^{2} \int_{0}^{c} x y^{2} d x\nonumber \\ &=-\alpha_{j}^{2} \int_{0}^{c} x J_{0}^{2}\left(\alpha_{j} x\right) d x .\label{eq:1} \end{align}
3. Noting that $$y(x)=J_{0}\left(\alpha_{j} x\right)$$, integrate the left hand side by parts and use the following to simplify the resulting equation.
1. $$J_{0}^{\prime}(x)=-J_{1}(x)$$ from Equation (5.5.5).
2. Equation (5.5.8).
3. $$J_{2}\left(\alpha_{j} c\right)+J_{0}\left(\alpha_{j} c\right)=0$$ from Equation (5.5.6).
4. Now you should have enough information to complete this part.
4. Use the results from parts b and c and Problem $$\PageIndex{16}$$ to derive the expansion coefficients for $x^{2}=\sum_{j=1}^{\infty} c_{j} J_{0}\left(\alpha_{j} x\right)\nonumber$ in order to obtain the desired expansion.

This page titled 5.7: Problems is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.