A COURSE OF MODERN ANALYSIS

4.5 Improper integrals. Principal values.

If \(\left|\,f(x) \,\right| \rightarrow \infty\) as \(x \rightarrow a+0\), then \(\lim\limits_{\delta \rightarrow +0} \displaystyle \int_{a+\delta}^b \!f(x)\, dx\) may exist, and is written simply \( \int_a^b f(x)\, dx\); this limit is called an improper integral.

If \(\left|\,f(x)\, \right| \rightarrow \infty\) as \(x \rightarrow c\), where \(a < c < b\), then \[\lim_{\delta \rightarrow +0} \int_a^{c-\delta} \!f(x)\, dx + \lim_{\delta' \rightarrow +0} \int_{c+\delta'}^b \! f(x)\,dx\] may exist; this is also written \( \int_a^b f(x)\, dx\), and is also called an improper integral; it might however happen that neither of these limits exists when \(\delta, \,\delta' \rightarrow 0\) independently, but \[\lim_{\delta \rightarrow +0} \left\{ \int_a^{c-\delta} \!f(x)\,dx+\int_{c+\delta}^b \! f(x)\,dx \right\} \] exists; this is called ‘Cauchy’s principal value of \( \int_a^b f(x)\, dx\)’ and is written for brevity \( P \int_a^b f(x)\, dx\).

Results similar to those of §§4.4–4.44 may be obtained for improper integrals. But all that is required in practice is (i) the idea of absolute convergence, (ii) the analogue of Bertrand’s test for convergence, (iii) the analogue of de la Vallée Poussin’s test for uniformity of convergence. The construction of these is left to the reader, as is also the consideration of integrals in which the integrand has an infinite limit at more than one point of the range of integration.[1]

Examples.

\[ \begin{align*} &(1) \quad \int_0^\pi \! x^{-\frac{1}{2}} \cos x\, dx \,\text{ is an improper integral.} \\ &(2) \quad \int^1 \! x^{\lambda-1} (1-x)^{\mu-1} dx \, \text{ is an improper integral if } 0 < \lambda < 1, \,0 < \mu <1. \\ & \qquad \qquad \text{It does not converge for negative values of }\lambda \text{ and } \mu .\\ &(3) \quad P \int_0^2 \!\frac{x^{\alpha-1}}{1-x} dx \,\text{ is the principal value of an improper integral}\\ & \qquad \qquad \text{when } 0 < \alpha <1. \end{align*} \]

4.51 The inversion of the order of integration of a certain repeated integral.

General conditions for the legitimacy of inverting the order of integration when the integrand is not continuous are difficult to obtain.

The following is a good example of the difficulties to be overcome in inverting the order of integration in a repeated improper integral.

Let \(f(x,y)\) be a continuous function of both variables, and let \(0 < \lambda \leq 1\), \(0 < \mu \leq 1\), \(0 < \nu \leq 1\); then \[ \begin{align*} &\int_0^1 \!dx \left\{\int_0^{1-x} \!x^{\lambda-1} y^{\mu-1} (1 -x-y)^{\nu-1} f(x,y) \, dy \right\} \\ & \qquad \qquad = \int_0^1 \!dy \left\{\int_0^{1-y} \! x^{\lambda-1} y^{\mu-1} (1-x-y)^{\nu-1} f(x,y) \, dx \right\}. \end{align*} \]

This integral, which was first employed by Dirichlet, is of importance in the theory of integral equations; the investigation which we shall give is due to W. A. Hurwitz.[2]

Let \(x^{\lambda-1} y^{\mu-1} (1 - x -y)^{\nu-1} f(x,y) = \phi(x,y)\); and let \(M\) be the upper bound of \(\left|\,f(x,y)\,\right|\).

Let \(\delta\) be any positive number less than \(\frac{1}{3}\).

Draw the triangle whose sides are \(x = \delta\), \(y = \delta\), \(x+y = 1- \delta\);[3] at all points on and inside this triangle \(\phi(x, y)\) is continuous, and hence, by §4.3 corollary, \[\int_\delta^{1-2\delta} \!dx \left\{\int_\delta^{1-x-\delta} \!\phi(x,y)\,dy \right\}=\int_\delta^{1-2\delta}\! dy \left\{\int_\delta^{1-y-\delta} \!\phi(x,y)\,dx \right\}.\]

Now \[ \begin{align*} &\int_\delta^{1-2\delta} \! dx \left\{\int_0^{1-x} \!\phi(x,y)\,dy \right\}\\ &\qquad =\int_\delta^{1-2\delta}\! dx \left\{\int_\delta^{1-x-\delta} \! \phi(x,y)\,dy \right\} + \int_\delta^{1-2\delta} \! I_1 \, dx + \int_\delta^{1-2\delta} \! I_2 \,dx, \end{align*} \] where \[I_1= \int_0^\delta \!\phi(x,y)\,dy, \quad I_2= \int_{1-x-\delta}^{1-x} \!\phi(x,y)\,dy. \]

But \[ \begin{align*} \left|\, I_1 \,\right|&\leq \int_0^\delta \! M x^{\lambda-1} y^{\mu-1} (1-x-y)^{\nu-1} dy\\ &\leq M x^{\lambda-1} (1-x-\delta)^{\nu-1} \int_0^\delta \! y^{\mu-1} dy, \end{align*} \] since \[ (1-x-y)^{\nu-1} \leq (1-x-\delta)^{\nu-1}. \]

Therefore, writing \(x = (1 - \delta)x_1\), we have[4] \[ \begin{align*} \left|\,\int_\delta^{1-2\delta} \! I_1 \, dx\, \right| & \leq M \delta^\mu \mu^{-1} \int_0^{1-\delta} \! x^{\lambda-1} (1-x-\delta)^{\nu-1} dx \\ & \leq M \delta^\mu \mu^{-1} (1-\delta)^{\lambda+\nu-1} \int_0^1 \! x_1^{\lambda-1} (1-x_1)^{\nu-1} dx_1 \\ & < M \delta^\mu \mu^{-1} (1-\delta)^{\lambda+\nu-1} B(\lambda,\nu) \rightarrow 0 \text{ as } \delta \rightarrow 0. \end{align*} \] The reader will prove similarly that \(\int_\delta^{1-2\delta}I_2\, dx \rightarrow 0\) as \(\delta \rightarrow 0\).

Hence[5] \[ \begin{align*} \int_0^1 \! dx \left\{\int_0^{1-x} \! \phi(x,y)\,dy \right\} & =\lim_{\delta \rightarrow 0}\int_\delta^{1-2\delta} \! dx \left\{\int_0^{1-x} \!\phi(x,y)\,dy \right\}\\ & =\lim_{\delta \rightarrow 0}\int_\delta^{1-2\delta} \!dx \left\{\int_\delta^{1-x-\delta} \!\phi(x,y)\,dy \right\}\\ & =\lim_{\delta \rightarrow 0}\int_\delta^{1-2\delta} \!dy \left\{\int_\delta^{1-y-\delta} \!\phi(x,y)\,dx \right\}. \end{align*} \]

by what has been already proved; but, by a precisely similar piece of work, the last integral is \[\int_0^1\! dy \left\{\int_0^{1-y} \!\phi(x,y)\,dx \right\}.\] We have consequently proved the theorem in question.

Corollary, Writing \(\xi = a + (b-a) x\), \(\eta = b-(b-a)y\), we see that, if \(\phi(\xi,\eta)\) is continuous, \[\begin{align*}&\int_a^b \!d\xi \left\{\int_\xi^b \!(\xi-a)^{\lambda-1} (b-\eta)^{\mu-1} (\eta-\xi)^{\nu-1} \phi(\xi,\eta)\,d\eta \right\}\\ &\qquad =\int_a^b \!d\eta \left\{\int_a^\eta \!(\xi-a)^{\lambda-1} (b-\eta)^{\mu-1} (\eta-\xi)^{\nu-1} \phi(\xi,\eta)\,d\xi \right\}\end{align*}\] This is called Dirichlet’s formula.

[Note. What are now called infinite and improper integrals were defined by Cauchy, Leçons sur le calc. inf. 1823, though the idea of infinite integrals seems to date from Maclaurin (1742). The test for convergence was employed by Chartier (1853). Stokes (1847) distinguished between ‘essentially’ (absolutely) and non-essentially convergent integrals though he did not give a formal definition. Such a definition was given by Dirichlet in 1854 and 1858 (see his Vorlesungen, 1904, p. 39). In the early part of the nineteenth century improper integrals received more attention than infinite integrals, probably because it was not fully realised that an infinite integral is really the limit of an integral.]

4.6 Complex integration.[6]

Integration with regard to a real variable \(x\) may be regarded as integration along a particular path (namely part of the real axis) in the Argand diagram. Let \(f(z)\), \((= P + iQ)\), be a function of a complex variable \(z\), which is continuous along a simple curve AB in the Argand diagram.

Let the equations of the curve be \[x = x (t),\quad y = y(t) \quad (a \leq t \leq b).\] Let \[x(a) + iy(a) = z_0 ,\quad x(b) + iy(b) = Z.\] Then if \(x(t),\, y(t)\) have continuous differential coefficients,[7] we define[8] \(\int_{z_{0}}^Z f(z)\, dz\) taken along the simple curve AB to mean \[\int_a^b \! (P+iQ)\left(\frac{dx}{dt}+i\frac{dy}{dt}\right) dt.\]

The ‘length’ of the curve AB will be defined as \( {\normalsize\int}_{a}^b \!\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dx}{dt}\right)^2} \,dt. \) It obviously exists if \(\frac{dx}{dt}\), \(\frac{dy}{dt}\) are continuous; we have thus reduced the discussion of a complex integral to the discussion of four real integrals, viz. \[\int_a^b \!P\frac{dx}{dt} dt, \quad \int_a^b \!P\frac{dy}{dt} dt, \quad \int_a^b \!Q\frac{dx}{dt} dt, \quad \int_a^b \!Q\frac{dy}{dt} dt.\]

By §4.13 example 4, this definition is consistent with the definition of an integral when AB happens to be part of the real axis.

Examples. \(\int_{z_{0}}^Z f(z)\, dz = -\int^{z_0}_Z f(z)\, dz \) the paths of integration being the same (but in opposite directions) in each integral.

\[ \begin{align*} &\int_{z_0}^Z \!dz = Z-z_0 \\ \\ &\int_{z_0}^Z \! z \, dz = \int_a^b\left\{x\frac{dx}{dt}-y\frac{dy}{dt}+i\left(x\frac{dy}{dt} + y\frac{dx}{dt}\right)\right\} dt \\ &\phantom{\int_{z_0}^Z \!z \, dz}= \left[\frac{1}{2}x^2 - \frac{1}{2}y^2 +ixy \right]_{t=a}^{t=b} = \frac{1}{2}(Z^2 - z_0^2). \end{align*} \]

4.61 The fundamental theorem of complex integration.

From §4.13, the reader will easily deduce the following theorem:

Let a sequence of points be taken on a simple curve \(z_0 Z\); and let the first \(n\) of them, rearranged in order of magnitude of their parameters, be called \(z_1^{\,(n)}, \, z_2^{\,(n)}, \, , \dots , z_n^{\,(n)}\) \((z_0^{\,(n)} = z_0, \, z_{n+1}^{\,(n)} = Z)\); let their parameters be \(t_1^{\,(n)}, \, t_2^{\,(n)}, \, , \dots , t_n^{\,(n)}\), and let the sequence be such that, given any number \(\delta\), we can find \(N\) such that, when \(n > N\), \(t_{r+1}^{\,(n)}-t_r^{\,(n)} < \delta \), for \(r= 0, 1, 2, \dots , n\); let \(\zeta_r^{\,(n)}\) be any point whose parameter lies between \(t_{r}^{\,(n)}, \,t_{r+1}^{\,(n)} \); then we can make \[\left|\, \sum_{r=0}^n \left(z_{r+1}^{\,(n)}- z_r^{\,(n)}\right) \, f\left(\zeta_r^{\,(n)}\right) - \int_{z_0}^Z \! f(z)\, dz \,\right|\] arbitrarily small by taking \(n\) sufficiently large.[9]

4.62 An upper limit to the value of a complex integral.

Let \(M\) be the upper bound of the continuous function \(\left|\,f(z)\,\right|\)

Then \[ \begin{align*} \left|\,\int_{z_0}^Z \! f(z)\,dz\,\right| &\leq \int_a^b \! \left|\,f(z)\,\right|\, \left|\left(\frac{dx}{dt} +i\frac{dy}{dt}\right)\right|\,dt \\ &\leq \int_a^b \!M \left\{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \right\}^{\frac{1}{2}} dt \\ \\ & \leq Ml, \end{align*} \] where \(l\) is the ‘length’ of the curve \(z_0 Z\).

That is to say, \(\displaystyle \left|\,\int_{z_0}^Z \! f(z)\,dz\,\right|\) cannot exceed \(Ml\).

4.7 Integration of infinite series.

We shall now shew that if \(S (z) = u_1(z) + u_2(z) + \cdots\) is a uniformly convergent series of continuous functions of \(z\), for values of \(z\) contained within some region, then the series \[\int_C \!u_1(z)\,dz + \int_C \!u_2(z)\,dz + \cdots ,\] (where all the integrals are taken along some path \(C\) in the region) is convergent, and has for sum \(\displaystyle \int_C \! S(z) \,dz\).

For, writing \[S (z) = u_1(z) + u_2(z) + \cdots + u_n(z) + R(z)\] we have \[\int_C \!S(z)\, dz = \int_C \!u_1(z)\,dz + \cdots +\int_C \!u_n(z)\,dz + \int_C \!R(z)\,dz.\]

Now since the series is uniformly convergent, to every positive number \(\epsilon\) there corresponds a number \(r\) independent of \(z\), such that when \(n \geq r\) have \(\left|\, R_n(z)\,\right| < \epsilon\), for all values of \(z\) in the region considered.

Therefore if \(l\) be the length of the path of integration, we have (§4.62) \[\left|\, \int_C \! R_n(z)\, dz \,\right| < \epsilon l .\]

Therefore the modulus of the difference between \(\displaystyle \int_C \!S(z)\, dz\) and \( \sum\limits_{m=1}^n \displaystyle \int_C \!u_m(z)\,dz\) can be made less than any positive number, by giving \(n\) any sufficiently large value. This proves both that the series \( \sum\limits_{m=1}^n \displaystyle \int_C \!u_m(z)\,dz\) convergent, and that its sum is \(\displaystyle \int_C \!S(z)\, dz\).

Corollary. As in §4.44 corollary, it may be shewn that[10] \[\frac{d}{dz}\sum_{n=0}^\infty u_n(z) = \sum_{n=0}^\infty \frac{d}{dz}u_n(z) \] if the series on the right converges uniformly and the series on the left is convergent.

Example 1. Consider the series \[\sum_{n=1}^\infty \frac{2x\{n(n+1) \sin^2 x^2-1\}\cos x^2}{\{1+n^2\sin^2 x^2\}\{1+(n+1)^2\sin^2x^2\}},\] in which \(x\) is real.

The nth term is \[\frac{2xn\cos x^2}{1+n^2\sin^2 x^2}-\frac{2x(n+1)\cos x^2}{1+(n+1)^2 \sin^2 x^2},\] and the sum of \(n\) terms is therefore \[\frac{2x\cos x^2}{1+\sin^2 x^2}-\frac{2x(n+1)\cos x^2}{1+(n+1)^2 \sin^2 x^2}.\]

Hence the series is absolutely convergent for all real values of \(x\) except \(\pm \sqrt{m\pi}\) where \(m=1, 2, \dots\); but \[R_n(x)=\frac{2x(n+1)\cos x^2}{1+(n+1)^2 \sin^2 x^2},\] and if \(n\) be any integer, by taking \(x=(n+1)^{-1}\) this has the limit 2 as \(n \rightarrow \infty\). The series is therefore non-uniformly convergent near \(x=0\).

Now the sum to infinity of the series is \(\displaystyle \frac{2x\cos x^2}{1+\sin^2 x^2}\), and so the integral from 0 to \(x\) of the sum of the series is \( \arctan\{\sin x^2\}\). On the other hand, the sum of the integrals from to \(x\) of the first \(n\) terms of the series is \[\arctan\{\sin x^2 \} - \arctan\{(n + 1) \sin x^2 \},\] and as \(n \rightarrow \infty\) this tends to \(\arctan\{\sin x^2 \} -\frac{1}{2}\pi\).

Therefore the integral of the sum of the series differs from the sum of the integrals of the terms by \(\frac{1}{2}\pi\).

Example 2. Discuss, in a similar manner, the series \[\sum_{n=1}^\infty \frac{2 e^n x\{1-n(e-1)+e^{n+1}x^2\}}{n(n+1)(1+e^nx^2)(1+e^{n+1}x^2)}\] for real values of \(x\).

Example 3. Discuss the series \[u_1+u_2+u_3+ \cdots ,\] where \[u_1=ze^{-z^2}, \quad u_n=nze^{-n z^2}-(n-1)ze^{-(n-1) z^2},\] for real values of \(z\).

The sum of the first \(n\) terms is \(nze^{-n z^2}\), so the sum to infinity is 0 for all real values of \(z\). Since the terms \(u_n\) are real and ultimately all of the same sign, the convergence is absolute.

In the series \[\int_0^z \!u_1 \,dz + \int_0^z \!u_2 \,dz + \int_0^z \!u_3 \,dz + \cdots ,\] the sum of \(n\) terms is \(\frac{1}{2}(1-e^{-nz^2})\), and this tends to the limit \(\frac{1}{2}\) as \(n\) tends to infinity; this is not equal to the integral from 0 to \(z\) of the sum of the series \(\sum u_n\).

The explanation of this discrepancy is to be found in the non-uniformity of the convergence near \(z = 0\), for the remainder after \(n\) terms in the series \(u_1 + u_2 + \cdots \) is \( -nze^{-nz^2}\); and by taking \(z=n^{-1}\) we can make this equal to \( -e^{-\left. 1\right/ n }\), which is not arbitrarily small; the series is therefore non-uniformly convergent near \(z = 0\).

Example 4. Compare the values of \[\int_0^z \!\left\{ \sum_{n=1}^\infty \!u_n \right\} dz \, \text{ and }\, \sum_{n=1}^\infty \int_0^z \! u_n \, dz , \] where \[u_n= \frac{2n^2z}{(1+n^2z^2) \log(n+1)} - \frac{2(n+1)^2z}{(1+(n+1)^2z^2) \log(n+2)}.\] (Trinity, 1903.)

REFERENCES.

Integration.

G. F. B. Riemann, Ges. Math. Werke, pp. 239–241.

P. G. Lejeune-Dirichlet, Vorlesungen. (Brunswick, 1904.)

G. F. Meyer, Bestimmte Integrale. (Leipzig, 1871.)

E. Goursat, Cours d’Analyse (Paris, 1910, 1911) Chs. iv, xiv.

C. J. de la Vallée Poussin, Cours d’Analyse Infinitésimale (Louvain and Paris, 1914), Ch. vi.

E. W. Hobson, Functions of a Real Variable (1907), Ch. v.

T. J. l’a. Bromwich, Theory of Infinite Series (1908), Appendix iii.