REFERENCES.
- E. Goursat, Cours d’ Analyse, ii. (Paris, 1911), Chs. xv, xvi.
- E. Borel, Leçons sur les séries divergentes (Paris, 1901).
- T. J. I’a. Bromwich, Theory of Infinite Series (1908), Chs. viii, x, xi.[1]
- Schlömilch, Compendium der höheren Analysis, ii., (Dresden, 1874).
Miscellaneous Examples.
If \(y - x - \phi(y) = 0\), where \(\phi\) is a given function of its argument, obtain the expansion \[ f(y) = f(x) + \sum_{m=1}^{\infty} \frac{1}{m!} \left\{ \phi(x) \right\}^{m} \left( \frac{1}{1-\phi'(x)} \frac{d}{d x}\!\! \right)^{m} f(x), \] where \(f\) denotes any analytic function of its argument, and discuss the range of its validity. \(\vphantom{\\ 3\\}\)
(Levi-Cività, Rend. dei Lincei, (5), xvi. (1907), p. 3.}Obtain (from the formula of Darboux or otherwise) the expansion \[ f(z) - f(a) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} (z-a)^{n}}{n!\:\! (1-r)^{n}} \left\{ f^{(n)}(z) - r^{n} f^{(n)}(a) \right\}; \] find the remainder after \(n\) terms, and discuss the convergence of the series.
Shew that \[\begin{align*} f(x+h) - f(x) &= \sum_{m=1}^{n} (-1)^{m-1} \frac{1 \cdot 3 \cdot 5 \cdots (2m-1)}{(m!)^{2}} \frac{h^{m}}{2^{m}} \left\{ f^{(m)}(x+h) - (-1)^{m} f^{(m)}(x) \right\} \\ & \quad + (-1)^{n} h^{n+1} \!\int_{0}^{1}\! \gamma_{n}(t)\:\! f^{(n+1)}(x + ht) \, d t, \end{align*}\] where \[\begin{align*} \gamma_{n}(t) & = \frac{x^{n+\frac{1}{2}} (1-x)^{n + \frac{1}{2}}}{(n!)^{2}} \frac{d^{n}}{d x^{n}}\!\! \left\{ x^{-\frac{1}{2}} (1-x)^{-\frac{1}{2}} \right\} \\ &= \frac{1}{\pi n!} \!\int_{0}^{1}\! (x-z)^{n} z^{-\frac{1}{2}} (1-z)^{-\frac{1}{2}} \, d z, \end{align*}\] and shew that \(\gamma_{n}(x)\) is the coefficient of \(n! t^{n}\) in the expansion of \(\left\{ (1-tx)(1+t-tx) \right\}^{-\frac{1}{2}}\) in ascending powers of \(t\).
By taking \[ \phi(x+1) = \frac{1}{n!} \left[ \frac{d^{n}}{d u^{n}}\!\! \left\{ \frac{(1-r)e^{xu}}{1 - r e^{-u}} \right\} \right]_{n=0} \] in the formula of Darboux, shew that \[\begin{align*} f(x+h) - f(x) =& - \sum_{m=1}^{n} a_{m} \frac{h^{m}}{m!} \left\{ f^{(m)}(x+h) - \frac{1}{r} f^{(m)}(x) \right\} \\ & \quad + (-1)^{n} h^{n+1} \!\int_{0}^{1}\! \phi(t) \:\! f^{(n+1)}(x+ht) \, d t, \end{align*}\] where \[ \frac{1-r}{1 - r e^{-u}} = 1 - a_{1} \frac{u}{1!} + a_{2} \frac{u^{2}}{2!} - a_{3} \frac{u^{3}}{3!} + \cdots. \]
Shew that \[\begin{align*} f(z) - f(a) &= \sum_{m=1}^{n} (-1)^{m-1} \frac{2 B_{m} (2^{2n} - 1)(z-a)^{2m-1}}{2m!} \left\{ f^{(2m-1)}(a) + f^{(2m-1)}(z) \right\} \\ & \quad + \frac{(z-a)^{2n+1}}{2n!} \!\int_{0}^{1}\! \psi_{2n}(t)\;\! f^{(2n+1)}\left\{ a + t(z-a) \right\} \, d t, \end{align*}\] where \[ \psi_{n}(t) = \frac{2}{n+1} \left[ \frac{d^{n+1}}{d u^{n+1}}\!\! \left( \frac{u e^{\:\! t\:\!u}}{e^{u} + 1} \right) \right]_{u=0}. \]
Prove that \[\begin{align*} & f(z_{2}) - f(z_{1}) = C_{1} (z_{2} - z_{1}) \;\! f'(z_{2}) + C_{2} (z_{2} - z_{1})^{2} f''(z_{1}) - C_{3} (z_{2} - z_{1})^{3} f'''(z_{2}) \\ & -C_{4} (z_{2}-z_{1})^{4} f^{(4)}(z_{1}) + \cdots + (-1)^{n} (z_{2} - z_{1})^{n+1} \!\int_{0}^{1}\! \left\{ \frac{d^{n}}{d u^{n}}\!\! \left( e^{\:\! t\:\! u} \mathrm{sech}\;\! u \right) \right\}_{\!\! u=0} \;\! f^{(n+1)}(z_{1} + t z_{2} - t z_{1}) \, d t; \end{align*}\] in the series plus signs and minus signs occur in pairs, and the last term before the integral is that involving \((z_{2}-z_{1})^{n}\), also \(C_{n}\) is the coefficient of \(z^{n}\) in the expansion of \(\cot\left( \frac{\pi}{4} - \frac{z}{2} \right)\) in ascending powers of \(z\). \(\vphantom{\\ 3\\}\)
(Trinity, 1899.)If \(x_{1}\) and \(x_{2}\) are integers, and \(\phi(z)\) is a function which is analytic and bounded for all values of \(z\) such that \(x_{1} \leq \mathfrak{Re}\:\!(z) \leq x_{2}\), shew (by integrating \[ \!\int \!\frac{\phi(z) \, d z}{ e^{\pm 2 \pi\:\! i\:\! z} - 1 } \] round indented rectangles whose corners are \(x_{1}\), \(x_{2}\), \(x_{2} \pm i\infty \), \(x_{1} \pm i\infty \)) that \[\begin{align*} & \frac{1}{2} \phi(x_{1})+\phi(x_{1} + 1) + \phi(x_{1} + 2) + \cdots + \phi(x_{2} - 1) + \frac{1}{2} \phi(x_{2}) \\ & \qquad = \!\int_{x_{1}}^{x_{2}}\! \phi(z) \, d z + \frac{1}{i} \!\int_{0}^{\infty}\! \frac{ \phi(x_{2}+iy) - \phi(x_{1}+iy) - \phi(x_{2}-iy) + \phi(x_{1}-iy)}{ e^{2 \pi y} - 1 } \, d y. \end{align*}\] Hence, by applying the theorem \[ 4n \!\int_{0}^{\infty}\! \frac{y^{2n-1}}{e^{2 \pi y} - 1} \, d y = B_{n}, \] where \(B_{1}, B_{2}, \ldots\) are Bernoulli’s numbers, shew that \[ \phi(1) + \phi(2) + \cdots + \phi(n) = C + \frac{1}{2} \phi(n) + \!\int^{n}\! \phi(z) \, d z + \sum_{r=1}^{\infty} \frac{(-1)^{r-1} B_{r}}{2r!} \phi^{(2r-1)}(n), \] (where \(C\) is a constant not involving \(n\)), provided that the last series converges. \(\vphantom{\\ 3\\}\)
(This important formula is due to Plana, Mem. della R. Accad. di Torino, xxv. (1820), pp. 403–418; a proof by means of contour integration was published by Kronecker, Journal für Math. cv. (1889), pp. 345–348. For a detailed history, see Lindelöf, Le Calcul des Résidus. Some applications of the formula are given in Chapter xii.)Obtain the expansion \[ u = \frac{x}{2} + \sum_{n=2}^{\infty} (-1)^{n-1} \frac{1 \cdot 3 \cdots (2n-3)}{n!} \frac{x^{n}}{2^{n}} \] for one root of the equation \(x = 2u + u^{2}\) and shew that it converges so long as \(\left|\, x \,\right| < 1\).
If \(S^{\:\! (m)}_{2n+1}\) denote the sum of all combinations of the numbers \[ 1^{2}, 3^{2}, 5^{2}, \ldots (2n-1)^{2}, \] taken \(m\) at a time, shew that \[ \frac{\cos z}{z} = \frac{1}{\sin z} +\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{(2n+2)!} \left\{ \frac{2^{2\:\! (n+1)}}{2n+3} - S^{\:\! (1)}_{2(n+1)} \frac{2^{2n}}{2n+1}+ \cdots + (-1)^{n} S^{\:\! (n)}_{2(n+1)} \frac{2^{2}}{3} \right\} \sin^{2n+1} z. \] (Teixeira.)
If the function \(f(z)\) is analytic in the interior of that one of the ovals whose equation is \(\left|\, \sin z \,\right| = C\) (where \(C \leq 1\)), which includes the origin, shew that \(f(z)\) can, for all points \(z\) within this oval, be expanded in the form \[\begin{align*} f(z) = f(0) &+ \sum_{n=1}^{\infty} \frac{ f^{(2n)}(0) + S^{\:\! (1)}_{2n} f^{(2n-2)}(0) + \cdots S^{\:\! (n-1)}_{2n} f''(0) }{2n!} \sin^{2n} z \\ &+\sum_{n=0}^{\infty} \frac{ f^{(2n+1)}(0) + S^{\:\! (1)}_{2n+1} f^{(2n-1)}(0) + \cdots + S^{\:\! (n)}_{2n+1} f'(0) }{(2n+1)!} \sin^{2n+1} z, \end{align*}\] where \(S^{\:\! (m)}_{2n}\) is the sum of all combinations of the numbers \[ 2^{2}, 4^{2}, 6^{2}, \ldots, (2n-2)^{2}, \] taken \(m\) at a time, and \(S^{\:\! (m)}_{2n+1}\) denotes the sum of all combinations of the numbers \[ 1^{2}, 3^{2}, 5^{2}, \ldots, (2n-1)^{2}, \] taken \(m\) at a time. \(\vphantom{\\ 3\\}\)
(Teixeira.)Shew that the two series \[ 2z + \frac{2 z^{3}}{3^{2}} + \frac{2 z^{5}}{5^{2}} + \cdots \] and \[ \frac{2z}{1 - z^{2}} - \frac{2}{1 \cdot 3^{2}} \left( \frac{2z}{1 - z^{2}} \right)^{3} + \frac{2 \cdot 4}{3 \cdot 5^{2}} \left( \frac{2z}{1 - z^{2}} \right)^{5} - \cdots \] represent the same function in a certain region of the \(z\) plane, and can be transformed into each other by Bürmann’s theorem. \(\vphantom{\\ 3\\}\)
(Kapteyn, Nieuw Archief, (2), iii. (1897), p. 225.)If a function \(f(z)\) is periodic, of period \(2 \pi\), and is analytic at all points in the infinite strip of the plane, included between the two branches of the curve \(\left|\, \sin z \,\right| = C\) (where \(C > 1\)), shew that at all points in the strip it can be expanded in an infinite series of the form \[\begin{align*} f(z) =& A_{0} + A_{1} \sin z + \cdots + A_{n} \sin^{n} z + \cdots \\ & \qquad + \cos z ( B_{1} + B_{2} \sin z + \cdots + B_{n} \sin^{n-1} z + \cdots ); \end{align*}\] and find the coefficients \(A_{n}\) and \(B_{n}\).
If \(\phi\) and \(f\) are connected by the equation \[ \phi(x) + \lambda f(x) = 0, \] of which one root is \(a\), shew that \[\begin{align*} F(x)= F &- \frac{\lambda}{1!}\frac{1}{\phi'}\:\! f\:\! F' \\ &+ \frac{\lambda^{2}}{1!\:\! 2!}\frac{1}{\phi'^{\:\! 3}} \left|\, \begin{array}{cc} \phi' & f^2 F' \\ \phi'' & \left( f^2 F' \right)' \end{array} \,\right| \\ &- \frac{\lambda^{3}}{1!\:\! 2!\:\! 3!}\frac{1}{\phi'^{\:\! 6}} \left|\, \begin{array}{ccc} \phi' & (\phi^2)' & f^3 F' \\ \phi'' & (\phi^2)'' &\left( f^3 F' \right)' \\ \phi''' & (\phi^2)''' &\left( f^3 F' \right)'' \end{array} \,\right| \\ \\ &+ \cdots , \end{align*}\] the general term being \( (-1)^{m} \dfrac{\lambda^{m}}{1!\:\! 2! \cdots m! (\phi')^{\frac{1}{2} m(m+1)}} \) multiplied by a determinant in which the elements of the first row are \(\phi'\), \((\phi^{2})'\), \((\phi^{3})'\), \(\ldots, (\phi^{m-1})'\), \((f^{m} F')\) and each row is the differential coefficient of the preceding one with respect to \(a\); and \(F\), \(f\), \(F', \ldots\) denote \(F(a)\), \(f(a)\), \(F'(a), \ldots\). \(\vphantom{\\ 3\\}\)
(Wronski, Philosophie de la Technie, Section ii. p. 381. For proofs of the theorem see Cayley, Quarterly Journal, xii. (1873), p. 221, Transon, Nouv Ann. de Math. xiii. (1874), p. 161, and C. Lagrange, Brux. Mém. Couronnés, 4°, xlvii. (1886), no. 2.)If the function \(W(a, b, x)\) be defined by the series \[ W(a,b,x) = x + \frac{a-b}{2!} x^{2} + \frac{(a-b)(a-2b)}{3!} x^{3} + \cdots, \] which converges so long as \[ \left|\, x \,\right| < \frac{1}{\left|\, b \,\right|}, \] shew that \[ \frac{d}{d x} W(a,b,x) = 1 + (a-b) W(a-b,b,x); \] and shew that if \[ y = W(a,b,x), \] then \[ x = W(b,a,y). \] Examples of this function are \[\begin{align*} W(1,0,x) &= e^{x} - 1, \\ \\ W(0,1,x) &= \log (1+x), \\ W(a,1,x) &= \frac{(1+x)^{a} - 1}{a}. \end{align*}\] (Ježek.)
Prove that \[ \frac{1}{ \sum_{n=0}^{\infty} a_{n} x^{n}}= \frac{1}{a_{0}} + \sum_{1}^{\infty} \frac{ (-1)^{n} x^{n} }{ n!\:\! a_{0}^{n+1} } G_{n}, \] where \[G_{n}=\left| \begin{array}{ccccccc} & 2a_1 & a_0 & \ 0 & 0 & \cdots & 0 & \\ & 4a_2 & 3a_1 & 2a_0 & 0 & \cdots & 0 & \\ & 6a_3 & 5a_2 & 4a_1 & 3a_0 & \cdots & 0 & \\ & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \\ & \mkern-15mu (2n-2)a_{n-1}\mkern-16mu & & \cdots & & \ddots & \mkern-10mu (n-1)a_0 \mkern-12mu & \\ & na_{n}&\mkern-14mu (n-1)a_{n-1}\mkern-16mu & & \cdots & & a_1 & \\ \end{array} \right| \] and obtain a similar expression for \[ \left\{ \sum_{n=0}^{\infty} a_{n} x^{n} \right\}^{\frac{1}{2}}. \] ( Mangeot, Ann. de l’Ecole norm. sup. (3), xiv (1897), p. 247.)
Shew that \[ \frac{1}{ \sum_{r=0}^{n} a_{r\:\!} x^{r} }= -\sum_{r=0}^{\infty} \frac{1}{r+1} \frac{\partial S_{r+1}}{\partial a_{1}} x^{r}, \] where \(S_{r}\) is the sum of the \(r\)-th powers of the reciprocals of the roots of the equation \[ \sum_{r=0}^{n} a_{r} x^{r} = 0. \] (Gambioli, Bologna Memorie, 1892.)
If \(f_{n}(z)\) denote the \(n\)th derivate of \(f(z)\), and if \(f_{-n}(z)\) denote that one of the \(n\)th integrals of \(f(z)\) which has an \(n\)-ple zero at \(z=0\), shew that if the series \[ \sum_{n=-\infty}^{\infty} f_{n}(z) g_{-n}(x) \] is convergent it represents a function of \(z + x\); and if the domain of convergence includes the origin in the \(x\)-plane, the series is equal to \[ \sum_{n=0}^{\infty} f_{-n}(z+x) g_{n}(0). \] Obtain Taylor’s series from this result, by putting \(g(z) = 1\). \(\vphantom{\\ 3\\}\)
(Guichard.)Shew that, if \(x\) be not an integer, \[ \underset{n \:\!\neq\:\! m}{\sum_{m=-\nu}^{\nu} \sum_{n=-\nu}^{\nu}} \frac{2x+m+n}{(x+m)^2(n+n)^2} \rightarrow 0 \] as \(\nu \rightarrow \infty\), provided that all terms for which \(m = n\) are omitted from the summation. \(\vphantom{\\ 3\\}\)
(Math. Trip. 1895.)Sum the series \[ \sum_{n=-q}^{p} \left( \frac{1}{(-1)^{n}\:\! x-a-n} + \frac{1}{n} \right), \] where the value \(n = 0\) is omitted, and \(p\), \(q\) are positive integers to be increased without limit. \(\vphantom{\\ 3\\}\)
(Math. Trip. 1896.)If \( F(x) = e^{\vphantom{\lim\limits_{=}}\int_{0}^{x} x \:\!\pi \cot (x \:\!\pi) \, d x} \), shew that \[ F(x) = e^{x} \frac{ \prod\limits_{n=1}^{\infty} \left\{ \left( 1 - \dfrac{x}{n} \right)^{n} e^{x \:\! + \frac{1}{2} \frac{x^{2}}{\!\!n}} \right\} }{ \prod\limits_{n=1}^{\infty} \left\{ \left( 1 + \dfrac{x}{n} \right)^{n} e^{-x \:\! + \frac{1}{2} \frac{x^{2}}{\!\!n}} \right\} } \] and that the function thus defined satisfies the relations \[ F(-x) = \frac{1}{F(x)}, \quad F(x) F(1-x) = 2 \sin x \pi. \] Further, if \[ \psi(z) = z + \frac{z^{2}}{2^{2}} + \frac{z^{3}}{3^{2}} + \cdots = - \!\int_{0}^{z}\! \log (1-t) \frac{\, d t}{t}, \] shew that \[ F(x) = e^{\frac{1}{2} \pi\:\! i\:\! x^{2} - \frac{1}{2 \pi\:\! i} \:\! \psi( 1 - e^{-2 \pi\:\! i\:\! x}\:\! ) } \] when \[ \left|\, 1 - e^{-2 \pi\:\! i\:\! x} \,\right| < 1. \] (Trinity, 1898.)
Shew that \[\begin{align*} & \left[ 1 + \left( \frac{k}{x} \right)^{n} \right] \left[ 1 + \left( \frac{k}{2 \pi - x} \right)^{n} \right] \left[ 1 + \left( \frac{k}{2 \pi + x} \right)^{n} \right] \left[ 1 + \left( \frac{k}{4 \pi - x}\right)^{n} \right] \left[ 1 + \left( \frac{k}{4 \pi + x} \right)^{n} \right] \cdots \\ & \qquad= \frac{ \prod\limits_{g=1}^{\lfloor \frac{1}{2} n \rfloor \vphantom{\lim\limits_{2}}} \sqrt{1 - 2 e^{-\alpha_{g}} \cos(x + \beta_{g}) + e^{-2\alpha_{g}}} \sqrt{1 - 2 e^{-\alpha_{g}} \cos(x - \beta_{g}) + e^{-2\alpha_{g}}} }{ 2^{\frac{1}{2} n} (1 - \cos x)^{\frac{1}{2} n} e^{-k \cos \left.\pi\:\!\middle/\:\! n\right.} }, \end{align*}\] where \[ \alpha_{g} = k \sin \frac{2g-1}{n} \pi, \quad \beta_{g} = k \cos \frac{2g-1}{n} \pi, \] and \[ 0 < x < 2 \pi. \] (Mildner.)
If \(\left|\, x\vphantom{z} \,\right| < 1\) and \(a\) is not a positive integer, shew that \[ \sum_{n=1}^{\infty} \frac{x^{n}}{n - a} = \frac{2 \pi i x^{a}}{1 - e^{2 a\:\! \pi\:\! i}} + \frac{x}{1 - e^{2 a\:\! \pi\:\! i}} \!\int_{C} \!\frac{t^{a-1} - x^{a-1}}{t - x} \, d t, \] where \(C\) is a contour in the plane enclosing the points \(0,x\). (Lerch, Casopis, xxi. (1892), pp. 65–68.)
If \(\phi_{1}(z)\), \(\phi_{2}(z), \ldots\) are any polynomials in \(z\), and if \(F(z)\) be any integrable function, and if \(\psi_{1}(z)\), \(\psi_{2}(z), \ldots\) be polynomials defined by the equations \[\begin{align*} & \!\int_{a}^{b}\! F(x) \frac{ \phi_{1}(z) - \phi_{1}(x) }{z - x} \, d x = \psi_{1}(z), \\ & \!\int_{a}^{b}\! F(x) \phi_{1}(x) \frac{ \phi_{2}(z) - \phi_{2}(x) }{z - x} \, d x = \psi_{2}(z), \\ & \phantom{\!\int_{a}^{b}\!} \cdots \\ & \!\int_{a}^{b}\! F(x) \phi_{1}(x) \phi_{2}(x) \cdots \phi_{m-1}(x) \frac{\phi_{m}(z) - \phi_{m}(x)}{z-x} \, d x = \psi_{m}(z), \end{align*}\] Shew that \[\begin{align*} & \!\int_{a}^{b}\! \frac{F(x) \, d x}{z - x} = \frac{\psi_{1}(z)}{\phi_{1}(z)} + \frac{\psi_{2}(z)}{\phi_{1}(z) \phi_{2}(z)} + \frac{\psi_{3}(z)}{\phi_{1}(z) \phi_{2}(z) \phi_{3}(z)} + \cdots \\ & + \frac{\psi_{m}(z)}{\phi_{1}(z) \phi_{2}(z) \cdots \phi_{m}(z)} + \frac{1}{\phi_{1}(z) \phi_{2}(z) \cdots \phi_{m}(z)} \!\int_{a}^{b}\! F(x) \phi_{1}(x) \phi_{2}(x) \cdots \phi_{m}(x) \frac{ \, d x }{z - x}. \end{align*}\]
A system of functions \(p_{0}(z)\), \(p_{1}(z)\), \(p_{2}(z), \ldots\) is defined by the equations \[ p_{0}(z) = 1, \quad p_{n+1}(z) = (z^{2} + a_{n} z + b_{n}) p_{n}(z), \] where \(a_{n}\) and \(b_{n}\) are given functions of \(n\), which tend respectively to the limits \(0\) and \(-1\) as \(n \rightarrow \infty\). Shew that the region of convergence of a series of the form \(\sum e_{n} p_{n}(z)\) where \(e_{1}, e_{2}, \ldots\) are independent of \(z\), is a Cassini’s oval with the foci \(+1, -1\).[2] Shew that every function \(f(z)\), which is analytic on and inside the oval, can, for points inside the oval, be expanded in a series \[ f(z) = \sum (c_{n} + z c'_{n}) p_{n}(z) % TODO: verify \] where \[ c_{n} = \frac{1}{2 \pi i} \!\int\! (a_{n}+z) q_{n}(z)\:\! f(x) \, d z, \quad c'_{n} = \frac{1}{2 \pi i} \!\int\! q_{n}(z)\:\! f(z) \, d z, \] the integrals being taken round the boundary of the region, and the functions \(q_{n}(z)\) being defined by the equations \[ q_{0} = \frac{1}{z^{2} + a_{0} z + b_{0}}, \quad q_{n+1}(z) = \frac{1}{z^{2} + a_{n+1} z + b_{n+1}} q_{n}(z). \] (Pincherle, Rend. dei Lincei, (4), v. (1889), p. 8.)
Let \(C\) be a contour enclosing the point \(a\), and let \(\phi(z)\) and \(f(z)\) be analytic when \(z\) is on or inside \(C\). Let \(\left|\, t \,\right|\) be so small that \[ \left|\, t\:\! \phi(z) \,\right| < \left|\, z - a \,\right| \] when \(z\) is on the periphery of \(C\). By expanding \[ \frac{1}{2 \pi i} \!\int_{C}\! f(z) \frac{1 - t\:\! \phi'(z)}{z - a - t\:\! \phi(z)} \, d z \] in ascending powers of \(t\), shew that it is equal to \[ f(a) + \sum_{n=1}^{\infty} \frac{ t^{n} }{n!} \frac{ d^{n-1} }{ d a^{n-1} }\!\! \left[ \:\! f'(a) \left\{ \phi(a) \right\}^{n} \right]. \] Hence, by using §6.3, §6.31, obtain Lagrange’s theorem.