Problems from Chapter 8 (The Normal Integral) of Irresistible Integrals by Boros and Moll

Here I present worked solutions of select problems from Chapter 8 (The Normal Integral) of Irresistible Integrals by Boros and Moll. This book has a plethora of interesting expositions and problems related to evaluating integrals. It also contains a lot of miscellaneous material that is either not related or only tangentially related to the thesis of the book. Thus I only chose problems of personal interest. Note that most of these problems can be found in Gradshteyn and Ryzhik (G & R), 5th Edition.

I will use the same designations of exercise and example as the authors.

We begin with the normal integral,
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x^{2}} \mathrm{d} x = \frac{\sqrt{\pi}}{2}
\label{eq:ii8-1}
\tag{1}
\end{equation}
Proof, begin with the gamma function
\begin{equation}
\Gamma(z) = \int\limits_{0}^{\infty} t^{z-1} \mathrm{e}^{-t} \mathrm{d} t
= 2\int\limits_{0}^{\infty} x^{2z-1} \mathrm{e}^{-x^{2}} \mathrm{d} x
\end{equation}
Where we let \(t=x^{2}\). Now let \(z=1/2\)
\begin{equation}
\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} = 2\int\limits_{0}^{\infty} \mathrm{e}^{-x^{2}} \mathrm{d} x
\end{equation}
rearrangement yields our result.

Exercise 8.1.2.a:

\begin{equation}
\int\limits_{0}^{\infty} x^{2n+1}\mathrm{e}^{-px^{2}} \mathrm{d} x = \frac{1}{2p^{\,n+1}}\int\limits_{0}^{\infty}z^{n}\mathrm{e}^{-z} \mathrm{d}z
= \frac{\Gamma(n+1)}{2p^{\,n+1}} = \frac{n!}{2p^{\,n+1}}
\end{equation}
We let \(z = px^{2}\) and \(n \in \mathbb{Z}_{0}^{+}\) and \(p \gt 0\).

Exercise 8.1.2.b: (G & R) 3.461.3

\begin{equation}
\int\limits_{0}^{\infty} x^{2n}\mathrm{e}^{-px^{2}} \mathrm{d} x = \frac{1}{2p^{\,n+1/2}}\int\limits_{0}^{\infty}z^{n-1/2}\mathrm{e}^{-z} \mathrm{d}z
= \frac{\Gamma(n+1/2)}{2p^{\,n+1/2}} = \frac{(2n!)\sqrt{\pi}}{2^{2n+1}n!p^{\,n+1/2}}
\end{equation}
with the same substitution.

Exercise 8.3.1: (G & R) 3.321.3

\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-a^{2}x^{2}} \mathrm{d} x
= \frac{1}{a}\int\limits_{0}^{\infty} \mathrm{e}^{-y^{2}} \mathrm{d} y
= \frac{\sqrt{\pi}}{2a}
\label{eq:ii8-2}
\tag{2}
\end{equation}
letting \(x=y/a\) and using equation \eqref{eq:ii8-1}.

Exercise 8.3.2: (G & R) 3.323.2

\begin{align}
\tag{a}
\int\limits_{-\infty}^{\infty} \mathrm{e}^{-a^{2}x^{2}+bx} \mathrm{d} x
&= \mathrm{e}^{b^{2}/4a^{2}} \int\limits_{-\infty}^{\infty}
\mathrm{e}^{-a^{2}(x-b/2a^{2})^{2}}\mathrm{d} x \\
\tag{b}
&= \mathrm{e}^{b^{2}/4a^{2}} \int\limits_{-\infty}^{\infty} \mathrm{e}^{-a^{2}z^{2}} \mathrm{d}z \\
&= 2\mathrm{e}^{b^{2}/4a^{2}} \int\limits_{0}^{\infty} \mathrm{e}^{-a^{2}z^{2}} \mathrm{d}z \\
&= \mathrm{e}^{b^{2}/4a^{2}} \frac{\sqrt{\pi}}{a}
\tag{c}
\end{align}
a. Complete the square.
b. Let \(z=x-b/2a^{2}\).
c. Use equation \eqref{eq:ii8-2}.

Exercise 8.3.3: (G & R) 3.361.2

\begin{equation}
\int\limits_{-a}^{\infty} \frac{\mathrm{e}^{-bx}}{\sqrt{x+a}}\mathrm{d} x
= \mathrm{e}^{ba} \int\limits_{0}^{\infty} z^{1/2} \mathrm{e}^{-bz}\mathrm{d} z
= \mathrm{e}^{ba} \Gamma\left(\frac{1}{2}\right)\frac{1}{\sqrt{b}}
= \mathrm{e}^{ba} \sqrt{\frac{\pi}{b}}
\end{equation}
we let \(x+a = z\).

Exercise 8.3.4: (G & R) 2.33

\begin{align}
\tag{a}
\int \mathrm{exp}(-(ax^{2}+2bx+c)) \mathrm{d} x
&= \mathrm{exp}\left(\frac{b^{2}}{a}-c\right) \int \mathrm{e}^{-a(x+b/a)^{2}} \mathrm{d} x \\
\tag{b}
& = \mathrm{exp}\left(\frac{b^{2}}{a}-c\right) \frac{1}{\sqrt{a}} \int \mathrm{e}^{-z^{2}} \mathrm{d}z \\
\tag{c}
&= \mathrm{exp}\left(\frac{b^{2}}{a}-c\right) \frac{1}{2} \sqrt{\frac{\pi}{a}} \,\mathrm{erf}(z) \\
&= \mathrm{exp}\left(\frac{b^{2}}{a}-c\right) \frac{1}{2} \sqrt{\frac{\pi}{a}} \,\mathrm{erf}\left(x\sqrt{a}+\frac{b}{\sqrt{a}}\right) + \mathrm{constant}
\label{eq:ii8-3}
\tag{3}
\end{align}
a. Complete the square.
b. Let \(z^{2} = a(x+b/a)^{2}\)
c. From the definition of the error function
\begin{equation}
\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int\limits_{0}^{x} \mathrm{exp}(-t^{2}) \mathrm{d} x
\end{equation}
we have
\begin{equation}
\int \mathrm{exp}(-t^{2}) \mathrm{d} x = \frac{\sqrt{\pi}}{2} \mathrm{erf}(x)
\end{equation}
Note that the book is missing the exponential term in the answer.

Exercise 8.3.5: (G & R) 3.462.7

The first step is to use equation \eqref{eq:ii8-3}, setting \(c=0\) and integrating from 0 to \(\infty\)
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{exp}(-(ax^{2}+2bx)) \mathrm{d} x
= \frac{1}{2}\sqrt{\frac{\pi}{a}} \,\mathrm{e}^{b^{2}/a} \Big[1-\mathrm{erf}(b/\sqrt{a})\Big]
\end{equation}
Now we differentiate both sides with respect to \(a\)
\begin{equation}
\int\limits_{0}^{\infty} x^{2}\mathrm{exp}(-(ax^{2}+2bx)) \mathrm{d} x
= -\frac{\sqrt{\pi}}{2} a^{-3/2} \mathrm{e}^{b^{2}/a} \Big[1-\mathrm{erf}(b/\sqrt{a})\Big] \left(\frac{1}{2}+\frac{b^{2}}{a} \right)
-\frac{b}{2a^{2}}
\end{equation}
this result does match that of Boros and Moll.

Exercise 8.3.6: (G & R) 3.468.2

\begin{align}
\tag{a}
\int\limits_{0}^{\infty} \frac{x\,\mathrm{exp}(-bx^{2})}{\sqrt{a^{2}+x^{2}}}\mathrm{d} x
&= \frac{1}{2\sqrt{b}} \int\limits_{0}^{\infty} \frac{\mathrm{e}^{-w}}{\sqrt{w+a^{2}b}}\mathrm{d}w \\
\tag{b}
&= \frac{\mathrm{exp}(a^{2}b)}{2\sqrt{b}} \int\limits_{a^{2}b}^{\infty} \mathrm{e}^{-z} z^{-1/2} \mathrm{d}z \\
\tag{c}
&= \frac{\mathrm{exp}(a^{2}b)}{2\sqrt{b}} \Gamma\left(\frac{1}{2},a^{2}b \right)
\end{align}
a. \(w=bx^{2}\)
b. \(z=w+a^{2}b\)
c.
\begin{equation}
\Gamma(a,x) = \int\limits_{x}^{\infty} t^{a-1} \mathrm{e}^{-t} \mathrm{d}t
\end{equation}
is the upper incomplete gamma function. We also have,
\begin{equation}
\mathrm{erf}(x) = \frac{1}{x}\sqrt{x^{2}} \left(1 -\frac{1}{\sqrt{\pi}}\Gamma\left(\frac{1}{2},x^{2} \right) \right)
\end{equation}
for \(x \gt 0\) and \(x \in \mathbb{R}\). So that
\begin{equation}
\Gamma\left(\frac{1}{2},x^{2} \right) = \sqrt{\pi}[1-\mathrm{erf}(x)]
\end{equation}
and our result matches that of the book.

Example 8.4.1: (G & R) 3.325

Prove
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}
\end{equation}
Let
\begin{equation}
\mathrm{L}(a,b) := \int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x
\label{eq:841-1}
\tag{1}
\end{equation}
Making the substitution \(t=x\sqrt{a}\) yields
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{\sqrt{a}} \int\limits_{0}^{\infty}
\mathrm{exp}(-t^{2}-\frac{ab}{t^{2}}) \mathrm{d} t
\label{eq:841-2}
\tag{2}
\end{equation}
Letting \(ab=c\) we call the integral in equation \eqref{eq:841-2} \(f(c)\),
\begin{equation}
f(c) = \int\limits_{0}^{\infty} \mathrm{exp}(-t^{2}-\frac{c}{t^{2}}) \mathrm{d} t
\label{eq:841-3}
\tag{3}
\end{equation}
so that
\begin{equation}
\mathrm{L}(a,b) = \frac{f(ab)}{\sqrt{a}}
\label{eq:841-4}
\tag{4}
\end{equation}
In equation \eqref{eq:841-3} we let \(y=\sqrt{c}/t\)
\begin{equation}
f(c) = \sqrt{c} \int\limits_{0}^{\infty} \mathrm{exp}(-y^{2}-\frac{c}{y^{2}}) y^{-2} \mathrm{d} y
\label{eq:841-5}
\tag{5}
\end{equation}

Combining equations \eqref{eq:841-3} and \eqref{eq:841-5}, we have
\begin{equation}
f(c) = \frac{1}{2} \int\limits_{0}^{\infty} \mathrm{exp}(-t^{2}-\frac{c}{t^{2}}) \left(1+\frac{\sqrt{c}}{t^{2}} \right) \mathrm{d} t
\label{eq:841-6}
\tag{6}
\end{equation}
Now we let \(s = t – \sqrt{c}/t\)
\begin{equation}
f(c) = \frac{\mathrm{e}^{- 2\sqrt{c}}}{2} \int\limits_{-\infty}^{\infty} \mathrm{exp}(-s^{2}) \mathrm{d} s = \frac{\sqrt{\pi}}{2} \mathrm{e}^{- 2\sqrt{c}} \lim_{z \to \infty} \mathrm{erf}(z) = \frac{\sqrt{\pi}}{2} \mathrm{e}^{- 2\sqrt{c}}
\label{eq:841-7}
\tag{7}
\end{equation}
Combining equations \eqref{eq:841-1}, \eqref{eq:841-4}, and \eqref{eq:841-7} yields our result.

I posted this solution and inquired about different solution methods at Mathematics Stack Exchange. There are two very interesting solutions there.

Exercise 8.4.1

\begin{align}
\label{a}
\int\limits_{0}^{\infty} \mathrm{e}^{-(ax+b/x)} \frac{1}{\sqrt{x}} \mathrm{d}x
&= 2\int\limits_{0}^{\infty} \mathrm{e}^{-(ay^{2}+b/y^{2})} \mathrm{d}y \\
\label{b}
&= \sqrt{\frac{\pi}{a}} \mathrm{e}^{-2\sqrt{ab}}
\end{align}
a. \(x=y^{2}\)
b. Use the result from Example 8.4.1 above.

Exercise 8.4.2.a

\begin{align}
\tag{a}
\int\limits_{0}^{\infty} \mathrm{exp}\left(-\mu x^{2} – \frac{a}{x^{2}}\right) – \mathrm{exp}(-\mu x^{2}) \mathrm{d}x
&= \frac{1}{2} \sqrt{\frac{\pi}{\mu}} \mathrm{e}^{-2\sqrt{a\mu}} \,-\, \frac{1}{\mu} \int\limits_{0}^{\infty} \mathrm{e}^{-z^{2}} \mathrm{d}z \\
\tag{b}
&= \frac{1}{2} \sqrt{\frac{\pi}{\mu}} \mathrm{e}^{-2\sqrt{a\mu}} \,-\, \frac{1}{2} \sqrt{\frac{\pi}{\mu}}
\end{align}
a. Use the result from Example 8.4.1 above.
b. The integral equals \(\mathrm{erf}(\infty) = 1\).

Exercise 8.4.2.b

Differentiate the result from Example 8.4.1 with respect to \(a\)
\begin{equation}
\int\limits_{0}^{\infty} x^{2} \, \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x
= \frac{\sqrt{\pi}}{2a} \mathrm{e}^{-2\sqrt{ab}} \left( \frac{1}{2\sqrt{a}} + \sqrt{b}\right)
\end{equation}

Exercise 8.4.2.c

Differentiate the result from Example 8.4.1 with respect to \(b\)
\begin{equation}
\int\limits_{0}^{\infty} x^{-2} \, \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x
= \frac{1}{2} \sqrt{\frac{\pi}{b}} \, \mathrm{e}^{-2\sqrt{ab}}
\end{equation}

Exercise 8.4.2.d

Differentiate the result from Exercise 8.4.2.c with respect to \(b\)
\begin{equation}
\int\limits_{0}^{\infty} x^{-4} \, \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x
= \frac{\sqrt{\pi}}{2b} \mathrm{e}^{-2\sqrt{ab}} \left( \frac{1}{2\sqrt{b}} + \sqrt{a} \right)
\end{equation}

Note that the problems in Exercise 8.4.2 are in G & R Section 3.472.

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