1. If $n$ is real and positive, and $|I(t)|,$ where $I(t)$ is the imaginary part of $t$, is less than either $n$ or $1$, we have

When $n=1$ the above formula reduces to if $t=0$, and then Similarly, if $\frac{1}{2} \sqrt{3}|I(t)|$ is less than either 1 or $n$, we have If in (4) we suppose $n=1$, we obtain and if $t=0$, and then In a similar manner we can prove that If we put $n=1$ in (7), we obtain Now Hence, dividing both sides of (7) by $t$, and making $t \to 0$, we obtain the result corresponding to (3) and (6), viz.: if then

2. I shall now shew that the integral (1) may be expressed in finite terms for all rational values of $n$. Consider the integral $$ J(t) = \int\limits^\infty_0 \frac{\cos tx}{\cosh \frac{1}{2} \pi x} \frac{ \ dx}{a^2 + x^2}. $$ If $R(a)$ and $t$ are positive, we have

and it is easy to see that this last equation remains true when $t$ is complex, provided $R(t)>0$ and $|I(t)|\leq \frac{1}{2} \pi$. Thus the integral $J(t)$ can be expressed in finite terms for all rational values of $a$. Thus, for example, we have and so on. Now let Then, if $R (a) > 0,$ Now let Then in virtue of (11); and therefore Now it is known that, if $\phi (n)$ is continuous and $$\int\limits^\infty_0 e^{-an} \phi (n) dn = 0, $$ for all positive values of $a$ (or even only for an infinity of such values in arithmetical progression), then $$ \phi (n) = 0,$$ for all positive values of $n$. Hence Equating the real and imaginary parts in (13) and (15) we have We can verify the results (18), (19) and (20) by means of the equation (1). This equation can be expressed as a functional equation in $F(n)$, nd it is easy to see that $f(n)$ satisfies the same equation. The right-hand side of these equations can be expressed in finite terms if $n$ is any rational number. For let $n=a/b,$ where $a$ and $b$ are any two positive integers and one of them is odd. Then the results (19) and (20) reduce to

Thus, for example, we have, when $a=1$ and $b=1$,

It is easy to verify that (23) and (24) satisfy the relation (2). The values of the integrals $$ \int\limits^\infty_0 \frac{\cos \pi n x^2}{\cosh \pi x} \ dx, \int\limits^\infty_0 \frac{\sin \pi n x^2}{\cosh \pi x} \ dx $$ can be obtained easily from the preceding results by putting $t=0$, and need no special discussion. By successive differentiations of the results (19) and (20) with respect to $t$ and $n$, we can evaluate the integrals for all rational values of $n$ and all positive integral values of $m$. Thus, for example, we have

3. We can get many interesting results by applying the theory of Cauchy's reciprocal functions to the preceding results. It is known that, if

then with the condition $\alpha \beta = 2 \pi/k $; with the condition $\alpha \beta = \pi/4k $; with the condition $\alpha \beta = \pi/6k,$ where 1, 5, 7, 11, 13, $\ldots$ are the odd natural numbers without the multiples of 3. There are of course corresponding results for the function such as $$ \alpha \{ \phi (\alpha) - \phi (3 \alpha) + \phi(5 \alpha) - \cdots \} = \Psi (\beta) - \Psi (3 \beta) + \Psi (5 \beta) - \cdots , $$ with the condition $\alpha \beta = \pi/2k.$ Thus from (23) and (27) (i) we obtain the following results. If

then $$ F(\alpha, \beta) = F(\beta, \alpha) = \sqrt{(2\alpha)} \{\frac{1}{2} + e^{-\pi \alpha} + e^{-4\pi\alpha} + e^{-9 \pi \alpha} + \cdots \}^2, $$ provided that $\alpha \beta = 1$.

4. If, instead of starting with the integral (11), we start with the corresponding sine integral, we can shew that, when $R(a)$ and $R(t)$ are positive and $|I(t)| \leq \pi $,

Hence the above integral can be expressed in finite terms for all rational values of $a$. For example, we have From (30) we can deduce that $R(t)$ being positive and $|I(t)|\leq \frac{1}{2} \pi$. The right-hand side can be expressed in finite terms for all rational values of $n$. Thus, for example, we have and so on. Applying the formula (28) to (33) and (34), we have, when $\alpha \beta = \frac{1}{4}$, By successive differentiation of (32) with respect to $t$ and $n$ we can evaluate the integrals for all rational values of $n$ and all positive integral values of $m$. Thus, for example, we have and so on. The denominators of the integrands in (25) and (36) are $\cosh \pi x$ and $\sinh \pi x$. Similar integrals having the denominators of their integrands equal to $$ \prod^r_1 \cosh \pi a_r x \sinh \pi b_r x$$ can be evaluated, if $a_r$ and $b_r$ are rational, by splitting up the integrand into partial fractions.

5. The preceding formulæ may be generalised. Thus it may be shewn that, if $R(a)$ and $R(t)$ are positive, $|I(t)|\leq \pi$, and $-1 \lt R(\theta) \lt 1 $, then

From (38) it can be deduced that, if $n$ and $R(t)$ are positive, $|I(t)|\leq \pi $, and $-1 \lt \theta \lt 1 $, then The right-hand side can be expressed in finite terms if $n$ and $\theta$ are rational. In particular, when $\theta = \frac{1}{3}$, we have where 1,2,4,5, $\ldots$ are the natural numbers without the multiples of 3. As an example, when $n=1$, we have

6. The formula (32) assumes a neat and elegant form when $t$ is changed to $t+\frac{1}{2} i \pi $. We have then

In particular, when $n=1$, we have We shall now consider an important special case of (42). It can easily be seen from (9) that the left-hand side of (42), when divided by $t$, tends to as $t \to 0$. But the limit of the right-hand side of (42) divided by $t$ can be found when $n$ is rational. Let then $n = a/b$, where $a$ and $b$ are any two positive integers, and let $$ \phi (n) = \int\limits^\infty_0 \frac{\cos \pi n x}{e^{2 \pi \sqrt{x}}-1} \ dx, \Psi (n) = \frac{1}{2 \pi n} + \int\limits^\infty_0 \frac{\sin \pi n x}{e^{2\pi \sqrt{x}}-1} \ dx.$$ The relation between $\phi(n)$ and $\Psi(n)$ has been stated already in (10'). From (42) and (44) it can easily be deduced that, if $a$ and $b$ are both odd, then It can easily be seen that these satisfy the relation (10'). Similarly, when one of $a$ and $b$ is odd and the other even, it can be shewn that where Thus, for example, we have and so on. By differentiating (42) with respect to $n$, we can evaluate the integrals for all rational values of $n$ and positive integral values of $m$. Thus, for example, we have and so on.