6.4 Connexion between the zeros of a function and the zeros of its derivative.
Macdonald has shewn[1] that if \(f(z)\) be a function of \(z\) analytic throughout the interior of a single closed contour \(C\), defined by the equation \(\left| \,f(z)\, \right| = M\), where \(M\) is a constant, then the number of zeros of \(f(z)\) in this region exceeds the number of zeros of the derived function \(f'(z)\) in the same region by unity.
On \(C\) let \(f(z) = M e^{i\theta}\); then at points on \(C\) \[ f'(z) = M e^{i\theta} i \frac{d \theta}{d z}, \quad f''(z) = M e^{i\theta} \left\{ i \frac{ d^{2}\! \theta }{ d z^{2} } - \left( \frac{d \theta}{d z} \right)^{2} \right\}. \]
Hence, by §6.31, the excess of the number of zeros of \(f(z)\) over the number of zeros of \(f'(z)\) inside \(C\:\!\)[2] is \[ \frac{1}{2\pi i} \int_{C} \frac{ f'(z) }{f(z)} \, dz - \frac{1}{2\pi i} \int_{C} \frac{ f''(z) }{ f'(z) } \, dz = -\frac{1}{2 \pi i} \int_{C} \left( \left. \frac{ d^{2}\! \theta }{ d z^{2} } \right/ \frac{d\theta}{d z} \right) \, dz. \] Let \(s\) be the arc of \(C\) measured from a fixed point and let \(\psi\) be the angle the tangent to \(C\) makes with \(Ox\); then \[\begin{align*} -\frac{1}{2 \pi i} \int_{C} \left( \left. \frac{ d^{2}\! \theta }{ d z^{2} } \right/ \frac{d\theta}{d z} \right) \, dz &= - \frac{1}{2 \pi i} \left[ \log \frac{d\theta}{d z} \right]_{C} \\ &= - \frac{1}{2 \pi i} \left[ \log \frac{d\theta}{d s} - \log \frac{d z}{d s} \right]_{C} . \end{align*}\] Now \(\log \dfrac{d \theta}{d s}\) is purely real and its initial value is the same as its final value; and \(\log \dfrac{d z}{d s} = i \psi\); hence the excess of the number of zeros of \(f(z)\) over the number of zeros of \(f'(z)\) is the change in \(\left.\psi\middle/2\pi\right.\) in describing the curve \(C\); and it is obvious[3] that if \(C\) is any ordinary curve, \(\psi\) increases by \(2\pi\) as the point of contact of the tangent describes the curve \(C\); this gives the required result.
Example 1. Deduce from Macdonald’s result the theorem that a polynomial of degree \(n\) has \(n\) zeros.
Example 2. Deduce from Macdonald’s result that if a function \(f(z)\), analytic for real values of \(z\), has all its coefficients real, and all its zeros real and different, then between two consecutive zeros of \(f(z)\) there is one zero and one only of \(f'(z)\).
REFERENCES.
- M. C. Jordan, Cours d’Analyse, ii. (Paris, 1894), Ch. vi.
- E. Goursat, Cours d’Analyse (Paris, 1911), Ch. xiv.
- E. Lindelöf, Le Calcul des Résidus (Paris, 1905), Ch. ii.
Miscellaneous Examples.
A function \(\phi(z)\) is zero when \(z=0\), and is real when \(z\) is real, and is analytic when \(\left| \,z\, \right| \leq 1\); if \(f(x,y)\) is the coefficient of \(i\) in \(\phi(x + iy)\), prove that if \(-1 < x < 1\), \[ \int_{0}^{2\pi}\! \frac{x \sin\theta }{ 1 - 2x\cos\theta + x^{2}}\, f(\cos\theta, \sin\theta) \, d\theta = \pi \:\! \phi(x). \] (Trinity, 1898.)
By integrating \(\dfrac{e^{\pm ai\:\!z}}{e^{2\pi\:\! z}-1}\) round a contour formed by the rectangle whose corners are \(0\), \(R\), \(R+i\), \(i\) (the rectangle being indented at \(0\) and \(i\)) and making \(R\rightarrow\infty\), shew that
\[ \int_{0}^{\infty}\! \frac{ \sin ax }{e^{2 \pi \:\! x} - 1} \, dx = \frac{1}{4} \frac{ e^{a} + 1 }{ e^{a} - 1 }- \frac{1}{2a}. \] (Legendre.)
By integrating \(\log (-z) Q(z)\) round the contour of §6.24, where \(Q(z)\) is a rational function such that \(z Q(z) \rightarrow 0\) as \(\left| \,z\, \right| \rightarrow 0\) and as \(\left| \,z\, \right| \rightarrow \infty\), shew that if \(Q(z)\) has no poles on the positive part of the real axis, \(\int_{0}^{\infty} Q(x) \, dx\) is equal to minus the sum of the residues of \(\log(-z) Q(z)\) at the poles of \(Q(z)\); where the imaginary part of \(\log(-z)\) lies between \(\pm\pi\).
Shew that, if \(a > 0, b > 0\), \[ \int_{0}^{\infty}\! e^{a \cos bx} \sin (a \sin bx) \frac{\, dx}{x} = \frac{1}{2} \pi \left( e^{a} - 1 \right). \]
Shew that \[ \int_{0}^{\frac{1}{2}\pi}\! \frac{a \sin 2x}{1 - 2a \cos 2x + a^{2}} x \, dx \, = \; \left\{\begin{array}{@{}ll@{}} \,\frac{1}{4} \pi \log (1+a), & (-1 < a < 1) \\ \\ \,\frac{1}{4} \pi \log (1+a^{-1}), & (a^{2} > 1). \\ \end{array}\right. \] (Cauchy.)
Shew that \[ \int_{0}^{\infty}\! \frac{\sin \phi_{1}x}{x} \frac{\sin \phi_{2}x}{x} \cdots \frac{\sin \phi_{n}x}{x} \cos a_{1} x \cdots \cos a_{m}\:\! x \frac{ \sin ax }{x} \, dx = \frac{\pi}{2} \phi_{1} \phi_{2} \cdots \phi_{n}, \] if \(\phi_{1}, \phi_{2}, \ldots \phi_{n}\), \(a_{1}, a_{2}, \ldots a_{m}\) be real and \(a\) be positive and \[ a > \left| \,\phi_{1}\, \right| + \left| \,\phi_{2}\, \right| +\cdots + \left| \,\phi_{n}\, \right|+ \left| \,a_{1}\, \right| + \cdots + \left| \,a_{m}\, \right|. \] (Stormer, Acta Math., xix. (1885), pp. 341–350.)
If a point \(z\) describes a circle \(C\) of centre \(a\), and if \(f(z)\) be analytic throughout \(C\) and its interior except at a number of poles inside \(C\), then the point \(u=f(z)\) will describe a closed curve \(\gamma\) in the \(u\)-plane. Shew that if to each element of \(\gamma\) be attributed a mass proportional to the corresponding element of \(C\), the centre of gravity of \(\gamma\) is the point \(r\), where \(r\) is the sum of the residues of \(\dfrac{f(z)}{z-a}\) at its poles in the interior of \(C\).\(\vphantom{\\ 3\\}\)
(Amigues, Nouv. Ann. de Math. (3), xii. (1893), pp. 142–148.)Shew that \[ \int_{-\infty}^{\infty}\! \frac{\, dx}{ (x^{2}+b^{2}) (x^{2}+a^{2})^{2} } = \frac{\pi (2a+b)}{ 2a^{3}b(a+b)^{2}}. \]
Shew that \[ \int_{0}^{\infty}\! \frac{\, dx}{(a+bx^{2})^{n}} = \frac{\pi}{ 2^{n} b^{\left.1\middle/2\right.} } \frac{1 \cdot 3 \cdots (2n-3)}{1 \cdot 2 \cdots (n-1)\vphantom{b^2}} \frac{1}{ a^{n \:\! -\:\!\left.1\middle/2\right.} }. \]
If \(F_{n}(z)= \prod\limits_{m=1}^{n-1} \prod\limits_{p=1}^{n-1} (1-z^{mp})\), shew that the series \[ f(z) = - \sum_{n=2}^{\infty} \frac{F_{n}(zn^{-1})}{ (z^{n}n^{-n} -1) n^{n-1} } \] is an analytic function when \(z\) is not a root of any of the equations \(z^{n} = n^{n}\); and that the sum of the residues of \(f(z)\) contained in the ring-shaped space included between two circles whose centres are at the origin, one having a small radius and the other having a radius between \(n\) and \(n + 1\), is equal to the number of prime numbers less than \(n + 1\).\(\vphantom{\\ 3\\}\)
(Laurent, Nouv. Ann. de Math. (3), xviii. (1899), pp. 234–241.)If \(A\) and \(B\) represent on the Argand diagram two given roots (real or imaginary) of the equation \(f(z) = 0\) of degree \(n\), with real or imaginary coefficients, shew that there is at least one root of the equation \(f'(z) = 0\) within a circle whose centre is the middle point of \(AB\) and whose radius is \(\frac{1}{2} AB \cot \dfrac{\pi}{n}\).\(\vphantom{\\ 3\\}\)
(Grace, Proc. Camb. Phil. Soc., xi. (1902), pp. 352–357.)Shew that, if \(0<\nu<1\), \[ \frac{e^{2\pi\:\! i \nu x}}{1 - e^{2\pi\:\! i x}} = \frac{1}{2\pi i} \lim_{n\rightarrow\infty} \sum_{k=-n}^{n} \frac{e^{2k\:\!\nu\pi\:\! i}}{k-x}. \] [Consider \(\displaystyle \int \frac{e^{(2\nu-1)\:\! z \pi\:\! i}}{\sin \pi z} \frac{\, dz}{z-x} \) round a circle of radius \(n+\frac{1}{2}\); and make \(n\rightarrow\infty\).]\(\vphantom{\\ 3\\}\)
(Kronecker, Journal für Math., cv. (1889), pp. 157–159, 345–354.)Shew that, if \(m > 0\), then \[\begin{align*} & \int_{0}^{\infty}\! \frac{\sin^{n} mt}{t^{\:\!n}} \, dt \\ = &\frac{\pi m^{n-1}}{2^{n} (n-1)!} \left( n^{n-1} - \frac{n}{1} (n-2)^{n-1} + \frac{n(n-1)}{2} (n-4)^{n-1} - \frac{n(n-1)(n-2)}{3!} (n-6)^{n-1} + \cdots \right). \end{align*}\] Discuss the discontinuity of the integral at \(m = 0\).
If \(A + B + C + \cdots = 0\) and \(a\), \(b\), \(c, \ldots\) are positive, shew that \[ \int_{0}^{\infty}\! \frac{A \cos ax + B \cos bx + \cdots + K \cos kx}{x} \, dx = -A \log a -B \log b - \cdots -K \log k. \] (Wolstenholme.)
By considering \( \displaystyle \int \frac{e^{x\:\! (k+ti)}}{k+t\:\! i} \, dt \) taken round a rectangle indented at the origin, shew that, if \(k > 0\), \[ i \lim_{\rho\rightarrow\infty} \int_{-\rho}^{\rho}\! \frac{e^{x\:\! (k+ti)}}{k+t i} \, dt = \pi i + \lim_{\rho\rightarrow\infty} P \int_{-\rho}^{\rho}\! \frac{e^{\:\! x t i}}{t} \, dt, \] and hence deduce, by using the contour of §6.222 example 2, or its reflexion in the real axis (according as \(x \geq 0\) or \(x < 0\)), that \[ \lim_{\rho\rightarrow\infty} \frac{1}{\pi} \int_{-\rho}^{\rho}\! \frac{e^{x\:\! (k+ti)}}{k+ti} \, dt = 2,\, , 1,\, \textrm{or } 0, \] according as \(x>0\), \(x = 0\), or \(x < 0\).
[This integral is known as Cauchy’s discontinuous factor.]Shew that, if \(\,0 < a < 2\), \(b > 0\), \(r > 0\), then \[ \int_{0}^{\infty}\! x^{a-1} \sin (\tfrac{1}{2} a\pi - bx ) \frac{r \, dx}{x^{2} + r^{2}} = \frac{1}{2} \pi r^{\:\! a-1} e^{-br}. \]
Let \(t>0\) and let \( \displaystyle \sum_{n=-\infty}^{\infty} e^{-n^{2} \pi\:\! t} = \psi(t). \)
By considering \( \displaystyle \int \! \frac{e^{-z^{2} \pi\:\! t}}{e^{2\pi\:\! i z} - 1} \, dz \) round a rectangle whose corners are \(\pm (N+\frac{1}{2}) \pm i\), where \(N\) is an integer, and making \(N \rightarrow \infty\), shew that \[ \psi(t) = \int_{-\infty - i}^{\infty - i} \frac{e^{-z^{2} \pi\:\! t}}{e^{2\pi\:\! i z} - 1} \, dz - \int_{-\infty + i}^{\infty + i} \frac{e^{-z^{2} \pi\:\! t}}{e^{2\pi\:\! i z} - 1} \, dz. \] By expanding these integrands in powers of \(e^{-2\pi\:\! i z}, e^{2\pi\:\! i z}\) respectively and integrating term-by-term, deduce from §6.22 example 3 that \[ \psi(t) = \frac{1}{ \sqrt{\pi t} } \psi(1/t) \int_{-\infty}^{\infty}\! e^{-x^{2}} \, dx. \] Hence, by putting \(t = 1\) shew that \[ \psi(t) = t^{-\frac{1}{2}} \psi(\left.1\middle/t\right.). \] (This result is due to Poisson, Journal de l’Ecole polytechnique, xii. (cahier xix), (1823), p. 420; see also Jacobi, Journal für Math., xxxvi. (1848), p. 109 [Ges. Werke, ii. (1882), p. 188].)Shew that, if \(t>0\), \[ \sum_{-\infty}^{\infty} e^{-n^{2} \pi t - 2 n \pi\:\! a t} = t^{-\frac{1}{2}} e^{\pi\:\! a^{2} t} \left\{ 1 + 2 \sum_{n=1}^{\infty} e^{-n^{2} \left.\pi\middle/t\right.} \cos 2n\pi a \right\}. \] (Poisson, Mem. de l’Acad. des Sci. vi. (1823), p. 592; Jacobi, Journal für Math., iii. (1828), pp. 403–404 [Ges. Werke, i. (1881), pp. 264–265]; and Landsberg, Journal für Math. cxi. (1893), pp. 234–253; see also §21.51.)