5.7: Problems

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. This is the standard Sturm-Liouville form for Bessel’s equation.
2. by considering

   $$\int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] d x .\nonumber$$

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.
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 $\alpha_{1}$.]
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: $\left(x y^{\prime}\right)^{\prime}=-\alpha_{j}^{2} x y .$
   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. in order to obtain the desired expansion.