Celebratio Mathematica

M. L. Cartwright: “The application of Abel’s method of summation to Dirichlet’s series,” J. Lond. Math. Soc. 3 : 4 (1928), pp. 262–267. MR 1574006 article

M. L. Cartwright: “On the relation between the functions represented by a power series and its associated Dirichlet’s series,” J. Lond. Math. Soc. 4 : 2 (1929), pp. 96–100. JFM 55.0201.02 article

M. L. Cartwright: “The zeros of certain integral functions,” Q. J. Math., Oxf. Ser. 1 : 1 (1930), pp. 38–59. JFM 56.0973.02 article

I propose to consider functions of the form \begin{equation*}\tag{1} f(z) = f(x+iy) = f(re^{i\theta}) = \int_a^b e^{it} \phi(t) \,dt, \end{equation*} where \( \phi(t) \) is a complex function, integrable in the sense of Lebesgue. The functions \begin{align*} U(z) &= \int_a^b \cos zt u(t)\,dt,\\ V(z) &= \int_a^b \sin zt v(t)\,dt \end{align*} may be reduced to the form (1) by simple transformations. Among them are many well-known functions such as Bessel functions.

The problem here is to determine approximately the total number of zeros, and the position of the zeros for which \( r \) is large.

My object here is to prove a number of theorems intermediate in generality between those of Titchmarsh and Hardy.

M. L. Cartwright: “On the relation between the different types of Abel summation,” Proc. London Math. Soc. (2) 31 : 1 (1930), pp. 81–96. MR 1577487 JFM 56.0908.01 article

Suppose that the series \( \sum_1^\infty a_n e^{-yn^{\nu}} \), where \( \nu > 0 \), is convergent for \( R(y) > 0 \), so that it defines an analytic function \( \phi_{\nu}(y) = \phi_{\nu}(re^{i\theta}) \) regular in the right half-plane; and suppose further that \[ \phi_{\nu}(y) \to l \] when \( y\to 0 \) so that \( -\alpha\leq\theta\leq\alpha \), where \( 0\leq\alpha < \frac{1}{2}\pi \); then the series \( \sum a_n \) is said to be summable \( (A_{\alpha},\nu) \) to sum \( l \). It is sufficient to assume that \( \phi_{\nu}(y)\to l \) when \( y\to 0 \) on \( \theta = \alpha \) and \( \theta = -\alpha \), and that \( \phi_{\nu}(y) \) is bounded in the angle \( -\alpha\leq\theta\leq\alpha \), for then, by a theorem due to Lindelöf [1909], \( \phi_{\nu}(y)\to l \) uniformly, when \( y\to 0 \) so that \( -\alpha\leq\theta\leq\alpha \).

The chief object of this paper is to show that if \( \sum a_n \) is summable \( (A_{\alpha},\nu) \), and if \( \sum a_n e^{-yn^{\mu}} \) is convergent for \( R(y) > 0 \), then \( \sum a_n \) is summable \( (A_{\beta}, \mu) \), provided that \[ \beta < \tfrac{1}{2}\pi - (\tfrac{1}{2}\pi - \alpha) \mu/\nu \] but I shall also prove several other theorems connecting the series \( \sum a_ne^{-yn^{\nu}} \) and \( \sum a_ne^{-yn^{\mu}} \).

M. L. Cartwright: “On the maximum modulus principle for functions with zeros and poles,” Proc. London Math. Soc. (2) 32 : 1 (1931), pp. 51–71. MR 1576004 JFM 56.0269.01 article

M. L. Cartwright: “On integral functions of integral order,” Proc. London Math. Soc. (2) 33 : 1 (1931), pp. 209–224. See also the similarly titled research report in Brit. Assoc. Rep. 99 (1932). MR 1576826 JFM 57.0362.01 Zbl 0003.21102 article

M. L. Cartwright: “The zeros of certain integral functions, II,” Q. J. Math., Oxf. Ser. 2 : 1 (1931), pp. 113–129. JFM 57.0361.03 Zbl 0002.19601 article

M. L. Cartwright: “The zeros of the cardinal function of interpolation,” J. London Math. Soc. 6 : 4 (1931), pp. 252–257. MR 1574626 JFM 57.0347.02 Zbl 0003.11002 article

M. L. Cartwright: “Integral functions of integral order,” Brit. Assoc. Rep. 99 (1932), pp. 341–342. See also similarly titled article in Proc. London Math. Soc. 33:1 (1931). JFM 58.0354.13 article

M. L. Cartwright: “Sur quelques propriétés des directions de Borel des fonctions entières d’ordre fini” [On some properties of directions of Borel of entire functions of finite order], C. R. Acad. Sci., Paris 194 (1932), pp. 2120–2122. JFM 58.0341.01 Zbl 0004.40303 article

M. L. Cartwright: “On functions regular in the unit circle,” pp. 47–48 in Verhandlungen des Internationalen Mathematiker-Kongresses Zürich 1932 [Proceedings of the International Congress of Mathematicians Zürich 1932] (Zürich, 5–12 September 1932), vol. 2. Edited by W. Saxner. Orell Füssli Verlag (Zürich and Leipzig), 1932. JFM 58.0353.16 incollection

M. L. Cartwright: “Sur les directions de Borel des fonctions entières d’ordre fini” [On the directions of Borel of entire functions of finite order], C. R. Acad. Sci., Paris 194 (1932), pp. 1889–1892. JFM 58.0340.02 Zbl 0004.40302 article

M. L. Cartwright: “Sur certaines fonctions entières d’ordre fini” [On certain entire functions of finite order], C. R. Acad. Sci., Paris 194 (1932), pp. 1718–1720. JFM 58.0340.01 Zbl 0004.40301 article

M. L. Cartwright: “Sur la rélation entre les directions de Borel de certaines fonctions entières et les singularites des fonctions analytiques” [On the relation between the directions of Borel of certain entire functions and the singularities of analytic functions], C. R. Acad. Sci., Paris 194 (1932), pp. 2280–2282. JFM 58.0338.03 Zbl 0004.40304 article

L. S. Bosanquet and M. L. Cartwright: “On the Hölder and Cesàro means of an analytic function,” Math. Z. 37 : 1 (1933), pp. 170–192. MR 1545389 JFM 59.0319.01 Zbl 0007.16802 article

M. L. Cartwright: “On analytic functions regular in the unit circle,” Q. J. Math., Oxf. Ser. 4 : 1 (1933), pp. 246–257. JFM 59.0325.01 Zbl 0008.11803 article

L. S. Bosanquet and M. L. Cartwright: “Some Tauberian theorems,” Math. Z. 37 : 1 (1933), pp. 416–423. MR 1545404 JFM 59.0235.03 Zbl 0007.34502 article

M. L. Cartwright: “On the minimum modulus of integral functions,” Proc. Camb. Philos. Soc. 30 : 4 (October 1934), pp. 412–420. JFM 60.0262.03 Zbl 0010.12201 article

M. L. Cartwright: “Mayer’s method of solving the equation \( dz=P\,dx+Q\,dy \),” Math. Gaz. 18 : 228 (May 1934), pp. 105–107. JFM 60.1126.02 Zbl 0008.39205 article

M. L. Cartwright: “On analytic functions regular in the unit circle, II,” Q. J. Math., Oxf. Ser. 6 : 1 (1935), pp. 94–105. JFM 61.0308.02 Zbl 0011.35803 article

Let \( f(z) = f(re^{i\theta}) = u(r,\theta) + iv(r,\theta) \) be regular for \( |z| < 1 \), and let \( f(0) = 0 \). We write \begin{align*} M(r) &= \max_{0\leq\theta < 2\pi}|f(re^{i\theta})|,\\ M_p(r,u) &= \Bigl\{ \frac{1}{2\pi}\int_0^{2\pi}|u(r,\theta)|^p d\theta \Bigr\}^{1/p} \quad (p > 0), \end{align*} and \( u_{+}(r,\theta) = u(r,\theta) \) (\( u > 0 \)), \( u_{+}(r,\theta) = 0 \) otherwise. In an earlier paper [1933] I showed that, if \[ u_{+}(r,\theta) < A(1-r)^{-\alpha}\quad (r < 1;\ \alpha > 0), \] then, for \( r < 1 \), \begin{align*} M(r) & < K(\alpha)A(1-r)^{-1} \quad (\alpha < 1),\\ M(r) & < KA(1-r)^{-1}\Bigl( \log\frac{1}{1-r} \Bigr)^2 \quad (\alpha = 1),\\ M(r) & < K(\alpha)A(1-r)^{-\alpha} \quad (\alpha > 1). \end{align*} The chief object of this paper is to prove a similar result for \( M_p(r,u_{+}) \) and \( M_p(r,f) \), where \( p\geq 1 \).

M. L. Cartwright: “On functions which are regular and of finite order in an angle,” Proc. London Math. Soc. (2) 38 : 1 (1935), pp. 158–179. MR 1576310 JFM 60.0262.01 Zbl 0010.12104 article

M. L. Cartwright: “On the directions of Borel of functions which are regular and of finite order in an angle,” Proc. London Math. Soc. (2) 38 : 1 (1935), pp. 503–541. MR 1576331 JFM 61.0339.03 Zbl 0011.31105 article

M. L. Cartwright: “On the directions of Borel of analytic functions,” Proc. London Math. Soc. (2) 38 : 1 (1935), pp. 417–457. MR 1576326 JFM 61.0339.02 Zbl 0010.40407 article

M. L. Cartwright: “Some inequalities in the theory of functions,” Math. Ann. 111 : 1 (1935), pp. 98–118. MR 1512983 JFM 61.0351.02 Zbl 0010.36203 article

Suppose that \[ w(z) = a_0 + a_1 z + a_2 z^2 + \cdots \] is regular for \( |z| = r < 1 \). It was first proved by Koebe [1909] that if \( w(z) \) is schlicht in the unit circle, and if \( a_0 = 0 \), \( a_1 = 1 \), then \[ |w(z)| \leq \phi(r)\qquad (0 < r < 1), \] where \( \phi \) is a function of \( r \) only. It was afterwards proved by Bieberbach [1927] that under these conditions \[ |w(z)| \leq \frac{r}{(1-r)^2}, \] and further results were proved by Littlewood [1925] for mean values of \( |w(z)| \) and for \( |a_n| \). Bieberbach’s result is the best possible, as is shown by the function \[ w(z) = \frac{z}{(1-z)^2}, \] which takes every value, except those on the negative real axis between \( -1/4 \) and \( \infty \) once, and once only.

The object of this paper is to prove similar results for functions which only take values \( p \) times, by methods differing fundamentally from those of previous writers on the subject.

M. L. Cartwright: “Some generalizations of Montel’s theorem,” Proc. Camb. Philos. Soc. 31 : 1 (1935), pp. 26–30. JFM 61.0307.02 Zbl 0011.02908 article

M. L. Cartwright: “On certain integral functions of order 1 and mean type,” Proc. Camb. Philos. Soc. 31 : 3 (August 1935), pp. 347–350. Zbl 0012.17106 article

M. L. Cartwright: “On certain integral functions of order one,” Q. J. Math., Oxf. Ser. 7 : 1 (1936), pp. 46–55. JFM 62.0354.01 Zbl 0013.35802 article

M. L. Cartwright: “Some uniqueness theorems,” Proc. London Math. Soc. (2) 41 : 1 (1936), pp. 33–47. MR 1575454 JFM 62.0330.01 Zbl 0013.21202 article

Let \( \phi(z) \) be regular for \( |z| < 1 \), and \( \psi(z) \) for \( |z| > 1 \). Suppose that \( \psi(z) \) is continuous on \( |z| = 1 \), and that the boundary values of \( \phi(z) \) and \( \psi(z) \) coincide on a certain arc of \( |z| = 1 \). Then \( \psi(z) \) is the analytical continuation of \( \phi(z) \) outside \( |z|= 1 \), and if the boundary values coincide at all points on \( |z| = 1 \), the function is regular at all points on \( |z| = 1 \). Since \( \psi(z) \) is regular outside \( |z| = 1 \), we have a function which is regular everywhere, and therefore reduces to a constant. The object of this paper is to discuss certain cases in which the coincidence of the boundary values on an arc of \( |z| = 1 \) implies that \( \phi(z) = \psi(z) = \) a constant. The theorems were suggested by a theorem of Levinson, and some theorems of Mandelbrojt.

M. L. Cartwright: “On the asymptotic values of functions with a non-enumerable set of essential singularities,” J. London Math. Soc. 11 : 4 (1936), pp. 303–306. MR 1574930 JFM 62.0363.01 Zbl 0015.16502 article

A set is said to be of logarithmic measure zero if for every \( \varepsilon > 0 \) there exists a sequence of circles with radii \( \rho_1, \rho_2,\dots \), covering the set, such that \[ \sum_{\nu=1}^{\infty}[\log^+(1/\rho_{\nu})]^{-1} < \varepsilon \]

Suppose that \( f(z) \) is meromorphic in an open simply connected domain \( D \), except for a set \( E \) of essential singularities of logarithmic measure zero. If \( f(z) \neq a \) near an essential singularity \( Z \), then, for every \( \delta > 0 \), \( f(z)\to a \) as \( z \) tends to some singularity in \( |z - Z| < \delta \) along a continuous path.

In other words \( a \) is an asymptotic value of \( f(z) \) at some singularity in every circle \( |z-Z| < \delta \).

M. L. Cartwright: “On the behaviour of an analytic function in the neighbourhood of its essential singularities,” Math. Ann. 112 : 1 (1936), pp. 161–186. MR 1513045 JFM 62.0364.02 Zbl 0013.06902 article

Suppose that \( f(z) \) is a one-valued function, meromorphic in a connected region \( D \), and having an essential singularity at a point \( z=Z \) on the boundary of \( D \). We associate with \( Z \) three sets of values:

The cluster set \( C_D(Z) \) of \( f(z) \) at \( Z \). This is the set of values \( a \) such that \[ \lim_{n\to\infty}f(z_n) = a, \] where \( z_1 \), \( z_2,\dots \) is a sequence of points tending to \( Z \) inside \( D \);
The range of values \( R_D(Z) \) of \( f(z) \) at \( Z \). A value \( a \) belongs to \( R_D(Z) \) if, and only if, \( f(z) \) takes the value \( a \) an infinity of times near \( Z \) inside \( D \);
The convergence set \( \Gamma_D(Z) \) of \( f(z) \) at \( Z \). This is the set of all values \( a \) such that \[ \lim_{z\to Z}f(z) = a, \] as \( z\to Z \) along a Jordan arc lying inside \( D \) except for the end point at \( Z \).

If \( D \) is the unit circle, we write simply \( C(Z) \), \( R(Z) \), \( \Gamma(Z) \).

The object of this paper is to discuss the sets \( C_D(Z) \), \( R_D(Z) \), and \( \Gamma_D(Z) \) taken over a set of non-isolated essential singularities; in particular we shall consider the measure of the sets \( R_D(Z) \) and \( \Gamma_D(Z) \), and the relation between \( R_D(Z) \) and \( \Gamma_D(Z) \).

M. L. Cartwright: “The exceptional values of functions with a non-enumerable set of essential singularities,” Q. J. Math., Oxf. Ser. 8 : 1 (1937), pp. 303–307. JFM 63.0284.03 Zbl 0017.40804 article

Let \( f(z) \) be a function which is meromorphic in an open domain \( D \) except for a set \( E \) of essential singularities. The object of this paper is to prove the following theorem and to discuss its relation to other known theorems of a similar type.

The set \( E \) is of linear measure zero, then \( f(z) \) takes all valves except perhaps a set of plane measure zero near each point of \( E \).

It is easy to see that, if \( E \) is of positive linear measure, \( f(z) \) may omit a set of positive plane measure, in fact \( f(z) \) may omit a whole region of the complex plane. I shall discuss later how far the result can be improved when \( E \) is of linear measure zero. The proof depends on the theory of minimal slit regions, and the complete result obtained can only be stated in terms of slit regions.

M. L. Cartwright: “On analytic functions with non-isolated essential singularities,” pp. 72 in Comptes rendus du Congrès International des Mathématiciens Oslo, 1936 [Proceedings of the International Congress of Mathematicians Oslo, 1936] (Oslo, 13–18 July 1936), vol. 2. A. W. Brøggers Bogtrykkeri (Oslo), 1937. JFM 63.0285.02 incollection

M. L. Cartwright: “On functions bounded at the lattice points in an angle,” Proc. London Math. Soc. (2) 43 : 1 (1938), pp. 26–32. MR 1575417 JFM 63.0278.03 Zbl 0016.26501 article

M. L. Cartwright: “On the level curves of integral and meromorphic functions,” Proc. London Math. Soc. (2) 43 : 6 (1938), pp. 468–474. MR 1575655 JFM 63.0284.01 Zbl 0017.31501 article

Let \( f(z) \) be an integral function; any curve on which \( |f(z)| \) is constant is called a level curve of \( f(z) \). Some properties of level curves have been discussed by Pennycuick [1936], and in his paper he mentions a conjecture of J. M. Whittaker, viz.:

If the integral functions \( f \) and \( g \) are of constant modulus on a closed curve \( \Gamma \), then \[ f(z) = K\{g(z)\}^a \quad (a > 0). \]

Valiron [1937] has shown that this is true except when \begin{align*}\tag{1} f(z) &=K(w(z)-a)^{\mu}\Bigl(\frac{w(z)-b}{w(z)}\Bigr)^{\nu},\\ g(z) &= K^{\prime}(w(z)-a)^{\mu^{\prime}}\Bigl(\frac{w(z)-b}{w(z)}\Bigr)^{\nu^{\prime}}, \end{align*} where \( w(z) \) is an integral function without zeros, \( \mu \), \( \mu^{\prime} \), \( \nu \), \( \nu^{\prime} \) are positive integers (including zero), and \( K \) and \( K^{\prime} \) are constants.

I propose to give a different proof which gives a similar, but less striking, theorem for meromorphic functions at the same time. The method is more direct; it shows the significance of the function \( w(z) \) more clearly, and gives certain results about asymptotic values which are not quite obvious from (1).

M. L. Cartwright: “On level curves of integral functions,” Quart. J. Math., Oxford Ser. 11 : 1 (1940), pp. 277–290. MR 0003219 JFM 66.0351.02 Zbl 0025.16904 article

Let \( f(z) \) be an integral or meromorphic function; a curve on which \( |f(z)| \) is constant is called a level curve of \( f(z) \). Level curves, or branches of level curves, which extend continuously to infinity are sometimes called open level curves and the others closed level curves. Any closed level curve contains at least one simple closed curve as a sub-set; and a simple closed curve on which \( |f(z)| = M \) encloses a simply-connected domain \( D \) in which \( |f(z)| < M \). Further, \( f(z) \) takes every value \( \alpha \) for which \( |\alpha| < M \) at least once in \( D \).

The chief object of this paper is to obtain some results for open level curves of integral functions corresponding to those about integral functions which have a common closed level curve.

M. L. Cartwright: “Grace Chisholm Young,” J. London Math. Soc. 19 : 75 Part 3 (1944), pp. 185–192. MR 0013107 Zbl 0060.01816 article

M. L. Cartwright and J. E. Littlewood: “On non-linear differential equations of the second order, I,” J. London Math. Soc. 20 : 3 (1945), pp. 180–189. Part II published in Ann. Math. 48:2 (1947). MR 0016789 Zbl 0061.18903 article

M. L. Cartwright and J. E. Littlewood: “On non-linear differential equations of the second order, II,” Ann. Math. (2) 48 : 2 (April 1947), pp. 472–494. Part I published in J. London Math. Soc. 20:3 (1945). Addendum published in Ann. Math. 48:2 (1947), errata pubished in Ann. Math. 50:2 (1949). MR 0021190 article

M. L. Cartwright: “Forced oscillations in nearly sinusoidal systems,” J. Inst. Elec. Engrs. III 95 : 34 (March 1948), pp. 88–96. An abstract was published in J. Inst. Elec. Engrs. I 95:89 (1948). MR 0027395 article

A large class of radio circuits which are analytically equivalent to an oscillatory network in parallel with a non-linear negative resistance, are represented fairly accurately by the differential equation \[ \ddot{\nu}-(\alpha+\beta\nu-\gamma\nu^2)\dot{\nu}+\omega^2\nu = E\omega_1^2\sin\omega_1 t \] where \( \alpha/\omega \), \( \beta/\omega \) and \( \gamma/\omega \) are small. The behaviour of the solutions of this equation near resonance has been discussed by Appleton, van der Pol and others.

The paper contains a more complete discussion of the synchronized and quasi-periodic solutions near resonance, their phases, amplitudes and energy, and also the way in which one type of stable solution gives way to another as the parameters of the system vary, for instance as the electromotive force or detuning vary. It is shown that the phase and amplitude favourable to synchronization are prolonged just before synchronization. This agrees with Appleton’s experimental results. It is also found that hysteresis occurs. The decrease in energy with the decrease in detuning is explained by the fact that the phase favourable to synchronization is that which opposes the motion and is prolonged.

M. L. Cartwright: “Topological aspect of forced oscillations,” Research 1 (1948), pp. 601–606. MR 0026199 article

M. L. Cartwright and J. E. Littlewood: “Errata: ‘On non-linear differential equations of the second order, II’,” Ann. Math. (2) 49 : 4 (October 1948), pp. 1010. Errata for article published in Ann. Math. 48:2 (1947). MR 0026200 article

M. L. Cartwright: “Forced oscillations in nearly sinusoidal systems,” J. Inst. Elec. Engrs. I 95 : 89 (May 1948), pp. 223. Abstract for an article published in J. Inst. Elec. Engrs. III 95:34 (1948). article

M. L. Cartwright, E. T. Copson, and J. Greig: “Non-linear vibrations,” Adv. Sci. 6 : 21 (1949), pp. 64–75. MR 0030665 Zbl 0032.21804 article

J. Greig	Related
Edward Thomas Copson	Related

M. L. Cartwright and J. E. Littlewood: “Addendum to ‘On non-linear differential equations of the second order, II’,” Ann. Math. (2) 50 : 2 (April 1949), pp. 504–505. Addendum to article published in Ann. Math. 48:2 (1947). MR 0030078 Zbl 0038.25001 article

M. L. Cartwright: “On nonlinear differential equations of the second order, III,” Proc. Cambridge Philos. Soc. 45 : 4 (1949), pp. 495–501. MR 0032073 Zbl 0038.24905 article

M. L. Cartwright: “Forced oscillations in nonlinear systems,” pp. 149–241 in Contributions to the theory of nonlinear oscillations, vol. 1. Edited by S. Lefschetz. Annals of Mathematics Studies 20. Princeton University Press, 1950. MR 0035355 Zbl 0039.09901 incollection

M. L. Cartwright: “Forced oscillations in nonlinear systems,” J. Research Nat. Bur. Standards 45 (1950), pp. 514–518. MR 0042013 article

M. L. Cartwright and J. E. Littlewood: “Some fixed point theorems,” Ann. Math. (2) 54 : 1 (July 1951), pp. 1–37. With an appendix by H. D. Ursell. MR 0042690 Zbl 0058.38604, 0054.07101 article

J. E. Littlewood	Related
H. D. Ursell	Related

M. L. Cartwright: “Non-linear vibrations,” Pi Mu Epsilon J. 1 : 4 (April 1951), pp. 131–137. article

M. L. Cartwright: “Non-linear vibrations: A chapter in mathematical history,” Math. Gaz. 36 : 316 (1952), pp. 81–88. Presidential address to the Mathematical Association, 3 January 1952. MR 0046967 Zbl 0049.24607 article

E. F. Collingwood and M. L. Cartwright: “Boundary theorems for a function meromorphic in the unit circle,” Acta Math. 87 : 1 (1952), pp. 83–146. See also article in Proceedings of the International Congress of Mathematicians (1952). MR 0050010 Zbl 0046.08402 article

M. L. Cartwright: “Van der Pol’s equation for relaxation oscillations,” pp. 3–18 in Contributions to the theory of nonlinear oscillations, vol. 2. Edited by S. Lefschetz. Annals of Mathematics Studies 29. Princeton University Press, 1952. MR 0052617 Zbl 0048.06902 incollection

The equation \[ \ddot{x} - k(1-x^2)\dot{x}+x=0 \] with \( k \) large and positive has only one periodic solution, other than \( x=0 \), and this is of a type usually described as a relaxation oscillation (as opposed to a sinusoidal oscillation). It was discussed by van der Pol [1926] who obtained a graphical solution for \( k=10 \) and by le Corbeiller [1936] who, using Liénard’s method, showed that the period \[ 2T = 2k(3/2 - \log_e 2) + O(k) ,\] and the greatest height \( h = 2 + O(1) \) as \( k\to\infty \). Other authors [Flanders and Stoker 1946; Haag 1943, 1944; LaSalle 1949] have also discussed the equation, in particular Dorodnitsin [1947] has obtained an asympotic formula for \( T \) with smaller error terms but his analysis is difficult to follow.

This paper is based on the joint work of Professor J. E. Littlewood and myself, largely on work which was done before that contained in our other published papers on nonlinear differential equations. We shall show that as \( k\to\infty \) \begin{align*} T &= k(3/2 - \log_e 2) + \frac{3(\alpha+\beta)}{2k^{1/3}} + O\Bigl(\frac{1}{k^{1/3}}\Bigr)\\ h &= 2 + \frac{\alpha+\beta}{3k^{4/3}} + O\Bigl(\frac{1}{k^{4/3}}\Bigr), \end{align*} where \( \alpha \) and \( \beta \) are constants determined as follows: The equation \[ \eta_0\frac{d\eta_0}{d\xi} = 2\xi\eta_0 + 1 \] has one and only one solution \( \eta_0^*(\xi) \) such that \( \eta_0^*(\xi) \to 0 \) as \( \xi\to -\infty \). \[ \alpha = \eta_0^*(0),\qquad \beta = \int_0^{\infty}\frac{d\xi}{\eta_0^*(\xi)}. \]

J. E. Littlewood and M. L. Cartwright: “Some topological problems connected with forced oscillations,” pp. 429–430 in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 30 August–6 September 1950), vol. 1. American Mathematical Society (Providence, RI), 1952. incollection

E. F. Collingwood and M. L. Cartwright: “Boundary theorems for functions meromorphic in the unit circle,” pp. 390 in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 30 August–6 September 1950), vol. 1. American Mathematical Society (Providence, RI), 1952. See also article in Acta Math. 87:1 (1952). incollection

M. L. Cartwright: Specialization in education. Headley Brothers (Ashford, UK), 1954. The Winifred Mercier Memorial Lecture delivered at Whitelands College, Putney, 1 October 1954. booklet

M. L. Cartwright: The mathematical mind. Oxford University Press (London and New York), 1955. The James Bryce memorial lecture. See also article in Math. Spectrum 2:2 (1969–70). MR 0073503 Zbl 0064.24103 booklet

M. L. Cartwright: “On the stability of solutions of certain differential equations of the fourth order,” Quart. J. Mech. Appl. Math. 9 : 2 (1956), pp. 185–194. MR 0080221 Zbl 0071.30901 article

M. L. Cartwright: “Some aspects of the theory of non-linear vibrations,” pp. 71–76 in Proceedings of the International Congress of Mathematicians, 1954 (Amsterdam, 2–9 September 1954), vol. 3. Erven P. Noordhoff (Groningen), 1956. MR 0084670 Zbl 0073.11403 incollection

M. L. Cartwright: Integral functions. Cambridge tracts in mathematics and mathematical physics 44. Cambridge University Press, 1956. MR 0077622 Zbl 0075.05901 book

M. L. Cartwright: “On the decomposition of functions regular in a circle,” J. London Math. Soc. 34 : 4 (1959), pp. 454–456. MR 0150311 Zbl 0092.07003 article

Suppose that \( f(z) \) is regular for \( |z| < 1 \). Then \begin{equation*}\tag{1} f(z) = \frac{1}{2\pi i}\int_C \frac{f(\zeta)}{\zeta - z}d\zeta, \end{equation*} where \( C \) is a simple closed contour in \( |\zeta| < 1 \) containing the point \( z \) in its interior domain. If \( z \) lies outside \( C \), the integral in (1) is zero. Now suppose that \( C = C_1\cup C_2 \), where \( C_1 \) is a simple arc from \( a \) to \( b \), and \( C_2 \) a simple arc from \( b \) to \( a \). Then \[ f(z) = \frac{1}{2\pi i}\Bigl\{\int_{C_1} {+} \int_{C_2}\Bigr\}\frac{f(\zeta)}{\zeta - z}d\zeta = f_1(z)+f_2(z) .\] Since \( f(\zeta) \) is regular on \( C \) and \( 1/(\zeta-z) \) is a regular function of \( z \) and \( \zeta \) except for \( \zeta = z \), \( f_1(z) \) and \( f_2(z) \) are regular in the plane cut along \( C_1 \) and \( C_2 \) respectively. Nevertheless their sum is \( f(z) \), which may have singularities on and outside \( |z| = 1 \) and certainly has none on \( C_1 \) or \( C_2 \). The object of this paper is to investigate the nature of the functions \( f_1(z) \), \( f_2(z) \) obtained in this way.

M. L. Cartwright: “Some decomposition theorems for certain invariant continua and their minimal sets,” Fund. Math. 48 : 3 (1959/1960), pp. 229–250. MR 0126259 Zbl 0095.37502 article

M. L. Cartwright: “The application of Lyapunov’s second method to certain differential equations of the third order,” Rend. Sem. Mat. Univ. Politec. Torino 19 (1959/1960), pp. 37–40. MR 0124583 Zbl 0096.29702 article

M. L. Cartwright: “Balthazar van der Pol,” J. London Math. Soc. 35 : 3 (1960), pp. 367–376. MR 0123447 Zbl 0094.00513 article

M. L. Cartwright: “Reduction of systems of linear differential equations to Jordan normal form,” Ann. Mat. Pura Appl. (4) 51 : 1 (1960), pp. 147–160. MR 0120416 Zbl 0166.03703 article

M. L. Cartwright: “Sheila Scott Macintyre,” J. London Math. Soc. 36 : 1 (1961), pp. 254–256. MR 0123449 Zbl 0095.00501 article

M. L. Cartwright: “Almost periodic solutions of certain second order differential equations,” pp. 100–110 in Seminario matematico e fisico di Milano [Milan mathematics and physics seminar] (Milan, 29–30 March 1960), published as Rend. Sem. Mat. Fis. Milano 31 : 1 (1961). MR 0155047 Zbl 0102.30901 incollection

M. L. Cartwright: “The stability of dynamical systems” in Atti del 6o Congresso dell’Unione Matematica Italiana [Proceedings of the 6th Congress of the Union of Italian Mathematicians] (Naples, 11–16 September 1959). Cremonese (Rome), 1961. incollection

M. L. Cartwright and E. F. Collingwood: “The radial limits of functions meromorphic in a circular disc,” Math. Z. 76 : 1 (1961), pp. 404–410. MR 0130379 Zbl 0156.08302 article

M. L. Cartwright: “A generalization of Montel’s theorem,” J. London Math. Soc. 37 : 1 (1962), pp. 179–184. MR 0140692 Zbl 0137.05002 article

Let \( S \) denote the half strip, \( \alpha < x < \beta \) and \( y > 1 \), of the \( z = x + iy \) plane, and suppose that \( f(z) \) is regular and \( |f(z)| < M \) in \( S \) unless otherwise stated. Montel’s theorem for a half strip states that, if \( f(a+iy)\to l \) as \( y\to\infty \) for any fixed \( a \) such that \( a < \alpha < \beta \), then, for every \( \delta > 0 \), \( f(x+iy)\to l \) as \( y\to\infty \) uniformly for \( \alpha + \delta \leq x \leq \beta - \delta \). The result is false if \( f(z) \) is replaced by its modulus in both hypothesis and conclusion, but Hardy and Carleman [1922], and later Hardy, Ingham and Pólya [1928] obtained some results for the case in which \( |f(x+iy)| \) tends to a limit along two lines in \( S \). The object of this note is to show how the results of Hardy, Ingham and Pólya can be improved when \( |f| \) tends to a limit which is not very much less than \( M \).

M. L. Cartwright: “Almost periodic solutions of systems of two periodic equations,” pp. 256–263 in Analytic methods in the theory of non-linear vibrations, vol. I. Izdat. Akad. Nauk Ukrain. SSR (Kiev), 1963. MR 0157366 Zbl 0122.32803 incollection

M. L. Cartwright: “Almost periodic solutions of equations with periodic coefficients,” pp. 207–218 in Nonlinear problems (Madison, WI, 30 April–2 May 1962). Edited by R. E. Langer. University of Wisconsin (Madison, WI), 1963. MR 0186871 Zbl 0111.28602 incollection

M. L. Cartwright: “Edward Charles Titchmarsh,” J. London Math. Soc. 39 : 1 (1964), pp. 544–565. Also published in Biograph. Mem. of Fell. of the Roy. Soc. 10 (1964) and (in abbreviated form) in Oxford dictionary of national biography 61 (2004). MR 0175740 Zbl 0124.24509 article

M. L. Cartwright: “The zeros of a certain entire function,” J. Math. Anal. Appl. 9 : 3 (1964), pp. 341–347. MR 0171014 Zbl 0145.08201 article

M. L. Cartwright: “From non-linear oscillations to topological dynamics,” J. London Math. Soc. 39 : 1 (1964), pp. 193–201. MR 0160988 Zbl 0119.07402 article

M. L. Cartwright: “Edward Charles Titchmarsh,” Biograph. Mem. of Fell. of the Roy. Soc. 10 (November 1964), pp. 305–324. Also published in J. London Math. Soc. 39:1 (1964) and (in abbreviated form) in Oxford dictionary of national biography 61 (2004). article

M. L. Cartwright: “Equicontinuous mappings of plane minimal sets,” Proc. London Math. Soc. (3) 14A : 1 (1965), pp. 51–54. Dedicated to J. E. Littlewood on his 80th birthday. MR 0177395 Zbl 0129.38705 article

Let \( f \) be a homeomorphism of the plane which maps a compact plane set \( M \) on to itself in such a way that the iterates \( f^n \), \( n = 0 \), \( \pm 1 \), \( \pm 2,\dots \), form an equicontinuous family on \( M \). Suppose further that \( M \) is a minimal set, that is to say, closed and invariant and irreducible with respect to these properties. Then \( M \) is the orbit-closure of every point \( p \) of \( M \). It is well known that if \( M \) is zero-dimensional then \( f \) is isochronous (regularly almost periodic). Further, Hemmingsen [1954] has shown that if \( M \) is connected then \( M \) is a simple closed curve and \( f \) is an irrational rotation. It follows that if \( M \) has \( n \) components then each component is a simple closed curve and \( f^n \) is an irrational rotation on each. The object of this note is to show that these are the only possible types of minimal set in the plane on which the mapping has equicontinuous iterates. That is to say there are no minimal sets with an infinity of components other than zero-dimensional sets for which the mapping has equicontinuous iterates.

M. L. Cartwright: “Topological problems of nonlinear mechanics,” pp. 135–142 in 3rd conference on nonlinear oscillations (Berlin, 25–30 May 1964), published as Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 1 : 1–2. Issue edited by K. Schröder, R. Reissig, and G. Maess. 1965. MR 0194659 Zbl 0199.14402 incollection

Kurt Schröder	Related
Gerhard Maess	Related
Rolf Reissig	Related

M. L. Cartwright: “Jacques Hadamard,” J. London Math. Soc. 40 : 1 (1965), pp. 722–748. MR 0181556 Zbl 0127.24403 article

M. L. Cartwright: “Almost periodic flows and solutions of differential equations,” Math. Nachr. 32 : 5 (1966), pp. 257–261. Lecture given at the celebrations of the 150th anniversary of Weierstrass’ birthday, 19–23 October 1963. An expanded version of this article was published in Proc. London Math. Soc. 17:2 (1967). MR 0224932 Zbl 0264.34057 article

M. L. Cartwright: “Almost periodic flows and solutions of differential equations,” Proc. London Math. Soc. (3) 17 : 2 (1967), pp. 355–380. Corrigenda published in Proc. London Math. Soc. 17:4 (1967). Expanded version of article published in Math. Nachr. 32:5 (1966). MR 1576613 Zbl 0155.41901 article

M. L. Cartwright: “Corrigenda: ‘Almost periodic flows and solutions of differential equations’,” Proc. London Math. Soc. (3) 17 : 4 (1967), pp. 768. Corrigenda for article published in Proc. London Math. Soc. 17:2 (1967). MR 1577210 article

M. L. Cartwright: “Almost periodic differential equations and almost periodic flows,” J. Differ. Equations 5 : 1 (1969), pp. 167–181. MR 0239191 Zbl 0167.07804 article

In a earlier paper [1967] I discussed the basic frequencies of uniformly almost periodic (u.a.p.) solutions of the systems \begin{align*} \mathbf{\dot{x}} &= \mathbf{\psi}(\mathbf{x}), & \mathbf{\dot{x}} &= \frac{d\mathbf{x}}{dt},\\ \mathbf{\dot{x}} &= \mathbf{\psi}(\mathbf{x},t), & \mathbf{\psi}(\mathbf{x},t) &= \mathbf{\psi}(\mathbf{x},t+2\pi), \end{align*} where \( \mathbf{x} \) and \( \mathbf{\psi} \) are vectors in \( R^n \), real Euclidean space of \( n \) dimensions. The object of this paper is to extend those results to the system \[\mathbf{\dot{x}} = \mathbf{\psi}(\mathbf{x},t), \qquad\mathbf{\psi}(\mathbf{x},t)\text{ u.a.p. in } t, \] where the almost periodicity is uniform with respect to \( \mathbf{x} \) for \( \mathbf{x} \) bounded.

M. L. Cartwright: “Limits of harmonic functions in a half-strip,” Bull. London Math. Soc. 1 : 1 (1969), pp. 40–42. MR 0240322 Zbl 0176.41103 article

M. L. Cartwright: “Comparison theorems for almost periodic functions,” J. London Math. Soc. (2) 1 : 1 (1969), pp. 11–19. MR 0288512 Zbl 0194.11504 article

M. L. Cartwright: “The mathematical mind,” Math. Spectrum 2 : 2 (1969–70), pp. 37–44. The material in this article formed the substance of the author’s James Bryce Memorial Lecture (1955). article

M. L. Cartwright: “Almost periodic solutions of differential equations and their basic frequencies,” pp. 205–207 in Proceedings of the fifth international conference on non-linear oscillations (Kiev, 25 August–4 September 1969), Section 2: Qualitative methods in the theory of nonlinear oscillations. Edited by Y. Mitropol’skii. Izdat. Akad. Nauk Ukrain. S.S.R. (Kiev), 1970. Zbl 0229.34042 incollection

M. L. Cartwright: “Mathematics and thinking mathematically,” Amer. Math. Mon. 77 : 1 (January 1970), pp. 20–28. MR 0255336 article

M. L. Cartwright: “Almost periodic solutions of differential equations and flows,” pp. 35–43 in Global differentiable dynamics (Case Western Reserve University, Cleveland, OH, 2–6 June 1969). Edited by O. Hajek, A. J. Lohwater, and R. McCann. Lecture Notes in Mathematics 235. Springer (Berlin), 1971. MR 0492589 Zbl 0238.34078 incollection

Arthur John Lohwater	Related
Otomar Hajek	Related
Roger McCann	Related

M. L. Cartwright and W. K. Hayman: “Edward Foyle Collingwood,” Biogr. Mem. Fellows. R. Soc. 17 (November 1971), pp. 139–158. article

Edward Foyle Collingwood	Related
Walter K. Hayman	Related

M. L. Cartwright: “Almost periodic minimal sets,” pp. 139–142 in Proceedings of symposium on differential equations and dynamical systems (University of Warwick, UK, September 1968–August 1969). Edited by D. Chillingworth. Lecture Notes in Mathematics 206. Springer (Berlin), 1971. incollection

M. L. Cartwright: “Conditions for boundedness of systems of ordinary differential equations,” pp. 19–26 in Ordinary differential equations (Mathematics Research Center, Naval Research Laboratory, Washington, DC, 14–23 June 1971). Edited by L. Weiss. Academic Press (New York), 1972. MR 0432972 Zbl 0309.34020 incollection

G. H. Hardy: Collected papers of G. H. Hardy, vol. 5. Edited by I. W. Busbridge, M. L. Cartwright, E. F. Collingwood, H. Davenport, T. M. Flett, H. Heilbronn, A. E. Ingham, R. Rado, R. A. Rankin, W. W. Rogosinski, F. Smithies, E. C. Titchmarsh, and E. M. Wright. Clarendon Press (Oxford), 1972. MR 0497843 Zbl 0251.01013 book

Richard Rado	Related
Robert Alexander Rankin	Related
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Ted Titchmarsh	Related
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Werner Wolfgang Rogosinski	Related
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Thomas Muirhead Flett	Related
Harold Davenport	Related
Edward Maitland Wright	Related
Hans Arnold Heilbronn	Related
Albert Edward Ingham	Related

G. H. Hardy: Collected papers of G. H. Hardy, vol. 6. Edited by L. S. Bosanquet, I. W. Busbridge, M. L. Cartwright, E. F. Collingwood, H. Davenport, H. Heilbronn, A. E. Ingham, R. Rado, R. A. Rankin, W. W. Rogosinski, F. Smithies, E. C. Titchmarsh, and E. M. Wright. Clarendon Press (Oxford), 1974. MR 0497844 Zbl 0323.01034 book

Richard Rado	Related
Robert Alexander Rankin	Related
Godfrey Harold Hardy	Related
Lancelot Stephen Bosanquet	Related
Ida Winifred Busbridge	Related
Ted Titchmarsh	Related
Werner Wolfgang Rogosinski	Related
Edward Foyle Collingwood	Related
Frank Smithies	Related
Harold Davenport	Related
Edward Maitland Wright	Related
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M. L. Cartwright and H. P. F. Swinnerton-Dyer: “Boundedness theorems for some second order differential equations, I,” Ann. Polon. Math. 29 (1974), pp. 233–258. Collection of articles dedicated to the memory of Tadeusz Ważewski, III. Part IV was published in Russian in Differ. Uravn. 14:11 (1978) and in English in Differ. Equations 14 (1979). MR 0355191 Zbl 0292.34023 article

Henry Peter Francis Swinnerton-Dyer	Related
Tadeusz Ważewski, III	Related

M. L. Cartwright: “Boundedness of solutions of second order differential equations,” pp. 67–70 in Topics in analysis (Jyväskylä, Finland, 1970). Edited by O. Lehto, I. S. Louhivaara, and R. Nevanlinna. Lecture Notes in Mathematics 419. Springer (Berlin), 1974. MR 0470329 Zbl 0296.34025 incollection

Rolf Herman Nevanlinna	Related
Ilppo Simo Louhivaara	Related
Olli Lehto	Related

M. Cartwright: “Some points in the history of the theory of nonlinear oscillations,” Bull. Inst. Math. Appl. 10 : 9–10 (1974), pp. 329–333. MR 0532224 article

M. L. Cartwright: “The Fatou asymptotic values of a function meromorphic in the unit disc and those of its derivative,” Bull. London Math. Soc. 7 : 1 (1975), pp. 61–64. MR 0369700 Zbl 0305.30032 article

M. Cartwright: “Some exciting mathematical episodes involving J. E. L.,” pp. 201–202 in Papers presented at the Symposium on Excitement in Mathematics (Cambridge, 1975), published as Bull. Inst. Math. Appl. 12 : 7 (1976). MR 0532531 incollection

M. L. Cartwright and H. P. F. Swinnerton-Dyer: “The boundedness of solutions of systems of differential equations,” pp. 121–130 in Differential equations (Keszthely, Hungary, 2–6 September 1974). Edited by M. Farkas. Colloquia Mathematica Societatis János Bolyai 15. North-Holland (Amsterdam), 1977. MR 0473332 Zbl 0361.34028 incollection

Henry Peter Francis Swinnerton-Dyer	Related
Miklós Farkas	Related

M. L. Cartwright: “A comparison of relationships between various kinds of damping and restoring forces,” pp. 93–101 in Problemy asimptoticheskoĭ teorii nelineĭnykh kolebaniĭ [Problems of the asymptotic theory of nonlinear oscillations]. Edited by V. S. Koroliuk. Naukova Dumka (Kiev), 1977. In Russian. MR 0594624 Zbl 0497.34025 incollection

M. L. Kartraĭt and H. P. F. Svinnerton-Daĭer: “Boundedness theorems for some second-order differential equations, IV,” Differ. Uravn. 14 : 11 (1978), pp. 1941–1979. In Russian. An English translation was published in Differ. Equations 14 (1979). MR 516689 article

M. L. Cartwright: “J. E. Littlewood,” Nature 271 : 5641 (1978), pp. 193. article

M. Cartwright: “John Edensor Littlewood,” Bull. Inst. Math. Appl. 14 : 4 (1978), pp. 87–90. MR 0497782 article

G. H. Hardy: Collected papers of G. H. Hardy, vol. 7. Edited by L. S. Bosanquet, I. W. Busbridge, M. L. Cartwright, E. F. Collingwood, H. Davenport, T. M. Flett, H. Heilbronn, A. E. Ingham, R. Rado, R. A. Rankin, W. W. Rogosinski, F. Smithies, E. C. Titchmarsh, and E. M. Wright. Clarendon Press (Oxford), 1979. MR 527275 Zbl 0396.01024 book

Richard Rado	Related
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Frank Smithies	Related
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M. L. Cartwight and H. P. F. Swinnerton-Dyer: “Boundedness theorems for some second-order differential equations, IV,” Differ. Equations 14 (1979), pp. 1378–1406. English version of Russian original published in Differ. Uravn. 14:11 (1978). Part I was published in Ann. Polon. Math. 29 (1974). Zbl 0429.34037 article

M. L. Cartwright: “Some Hardy–Littlewood manuscripts,” Bull. London Math. Soc. 13 : 4 (1981), pp. 273–300. MR 620040 Zbl 0464.01004 article

J. E. Littlewood	Related
Godfrey Harold Hardy	Related

M. L. Cartwright: “Manuscripts of Hardy, Littlewood, Marcel Riesz and Titchmarsh,” Bull. London Math. Soc. 14 : 6 (1982), pp. 472–532. MR 679927 Zbl 0501.01009 article

Marcel Riesz	Related
Godfrey Harold Hardy	Related
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M. Cartwright: “Note on A. G. Cock’s paper ‘Chauvinism and internationalism in science’: The International Research Council, 1919–1926,” Notes and Records Roy. Soc. London 39 : 1 (1984), pp. 125–128. The article in question is Notes and Records Roy. Soc. London 37:2 (1983), pp. 249–288. MR 782425 Zbl 0592.01038 article

M. L. Cartwright: “An unsymmetrical van der Pol equation with stable subharmonics,” Int. J. Non-Linear Mech. 20 : 5–6 (1985), pp. 359–369. Zbl 0615.34035 article

M. L. Cartwright: “Later Hardy and Littlewood manuscripts,” Bull. London Math. Soc. 17 : 4 (1985), pp. 318–390. MR 806635 Zbl 0579.01009 article

J. E. Littlewood	Related
Godfrey Harold Hardy	Related

M. L. Cartwright and G. E. H. Reuter: “On periodic solutions of van der Pol’s equation with sinusoidal forcing term and large parameter,” J. London Math. Soc. (2) 36 : 1 (1987), pp. 102–114. MR 897678 Zbl 0641.34042 article

M. Cartwright: “Moments in a girl’s life,” Bull. Inst. Math. Appl. 25 : 3–4 (1989), pp. 63–67. MR 996106 article

N. P. Erugin: “Mary Lucy Cartwright,” Differ. Uravn. 25 : 9 (1989), pp. 1642–1646. MR 1024691 Zbl 0673.01013 article

S. L. McMurran and J. J. Tattersall: “The mathematical collaboration of M. L. Cartwright and J. E. Littlewood,” Amer. Math. Mon. 103 : 10 (1996), pp. 833–845. MR 1427114 Zbl 0887.01017 article

James J. Tattersall	Related
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S. McMurran and J. Tattersall: “Mary Cartwright (1900–1998),” Notices Amer. Math. Soc. 46 : 2 (1999), pp. 214–220. MR 1673764 Zbl 0912.01009 article

James J. Tattersall	Related
Shawnee L. McMurran	Related

J. Tattersall and S. McMurran: “An interview with Dame Mary L. Cartwright, D.B.E., F.R.S,” College Math. J. 32 : 4 (2001), pp. 242–254. MR 1860301 Zbl 0998.01551 article

James J. Tattersall	Related
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T. Körner: “Mary Cartwright,” Chapter 19, pp. 282–298 in Cambridge scientific minds. Edited by P. Harman and S. Mitton. Cambridge University Press, 2002. MR 2132156 incollection

Peter Michael Harman	Related
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W. K. Hayman: “Mary Lucy Cartwright (1900–1998),” Bull. London Math. Soc. 34 : 1 (2002), pp. 91–107. MR 1866432 Zbl 1032.01034 article

M. L. Cartwright: “Titchmarsh, Edward Charles (1899–1963)” in Oxford dictionary of national biography, new edition, vol. 61. Edited by H. C. G. Matthew and B. H. Harrison. Oxford University Press, 2004. Abbreviated version of obituary published in J. London Math. Soc. 39:1 (1964) and in Biograph. Mem. of Fell. of the Roy. Soc. 10 (1964). incollection

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Mary Lucy Cartwright

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