Felix Klein 1849 ~ Max Noether 1844 ~ Perkins ~ Colloquiu

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1 ~ 15 The Chinese Journal for the History of Science and Technology Vol. 35 No ~ ~ N092 O1-09 A William Fogg Osgood 1864 ~ Functions of Real Variables 1 Functions of a Complex Variable ~ 6 Joseph L. Walsh 1895 ~ 1973 Lehrbuch der Funktionentheorie

2 Felix Klein 1849 ~ Max Noether 1844 ~ Perkins ~ Colloquium Lecturer A First Course in the Differential and Integral Calculus ~ 514

3 ~ ~ George David Birkhoff 1884 ~

4 ~ ~ ~ ~ ~ ~ Cauchy Characteristics ⅲ 1935 Theory of Functions of Real Variables

5 Convergence of Infinite Series 2 The Number System lim u n + 1 /u n = 1 Kummer's Criterion 3 Point Sets Limits Continuity 4 Derivatives Integrals Implicit Functions 5 Uniform Convergence 6 The Elementary Functions M s n m 7 Algebraic Transformations of Infinite Series Series of Series 8 Fourier's Series 9 Definite Integrals Line Integrals 2 u / x y = 2 u / y x - μ x The de la Vallée-Poussin μ x -Test sinx dx x 0 Duhamel's Theorem Pdx + Qdy c 10 Γ The Gamma Function Γ Gauss's Product 11 Fourier's Integral 12 Differential Equations Existence Theorems Derived Integrals 1 ⅲ

6 ⅳ 1 ⅲ 1 Introduction to the Calculus Advanced Calculus s m x - ε < s n x < s m x + ε y = s m x - ε y = s m x + ε ε' < ε Bessel 3 1 Pierce Newtonian Potentials 1 2 William Elwood Byerly 1849 ~ 1935 Fourier' s Series and Spherical Harmonics 3 Andrew Gray 1847 ~ 1925 George Ballard Mathews 1861 ~ 1922 A Treatise on Bessel Functions and Their Applications to Physics Poisson 13 8 ~ 9 1 Benjamin Osgood Pierce 1854 ~ 1914 Elements of the Theory of the Newtonian Potential Function

7 ⅲ 1935 Riemann Cauchy David Raymond Curtiss 1878 ~ 1953 Analytic Functions of a Complex Variable 2 James Pierpont 1866 ~ 1938 Functions of a Complex Variable 3 douard-jean-baptiste Goursat 1858 ~ 1936 Earle Raymond Hedrick 1876 ~ 1943 A Course in Mathematical Analysis ~ 6 Cauchy Riemann Cauchy ~ 107 Cauchy Cauchy ⅲ Complex Numbers 2 Analytic Functions. Linear Transformations 3 Conformal Mapping w = z a w = sin - 1 z w = 1 z Kinematic Treatment

8 Riemann's Surfaces 5 The Cauchy Theory w = z a w 2 = G z w 3-3w = z Functions of the Double Pyramid Liouville's Theorem Morera's Theo- rem - The Cauchy-Taylor Development 6 The Further Development. Weierstrass Riemann Laurent's Theorem Darboux's Theorem Map of a 1 2 Rectangle on a Circle 7 Analytic Continuation 8 The Logarithmic Potential 3 The Theorem of Vanishing Flux 4 μ u = c 5 9 Conformal Existence of a Map of a Simply Connected Region Green's Function Riemann 5 Cauchy

9 A Century of Mathematics in America ~ ⅲ ~ ⅳ ~ ~ ~ ~ 95 M ~ ~ ~ ~ After Osgood's retirement from Harvard in 1933 he spent two years 1934 ~ 1936 teaching at the National University of Peking. Two books in English of his lectures there were prepared by his students and published there in 1936 Functions of Real Variables and Functions of a Complex Variable. Both books borrowed largely from the Funktionentheorie

10 Let a function f x be defined in the neighborhood of a point x = x 0. Form the difference-quotient 1 f x 0 + Δx - f x 0 Δx where x 0 + Δx is a point of the above neighborhood distinct from x 0. If the quotient approaches a limit as Δx approaches 0 the function is said to have a derivative or be differentiable at the point x 0. We write f x 0 + Δx - f x 0 2 lim = D Δx = 0 Δx x y = f' x 0. Ist x 0 ein innerer Punkt des Intervalls und kon- If the ratio 1 approaches a limit when Δx approaches 0 passing only through positive values f x is said to have a forward derivative. And gegen einen Grenzwert so definiert man letzvergiert Δy /Δx beim Grenzübergange limδx = 0 similarly for a backward derivative. If and only if these two are equal teren als die Ableitung der Funktion f x im will f x have a derivative in the point x 0. But if x 0 is an end point of Punkte x 0 und bezeichnet ihn mit f' x 0 the domain of definition of f x then f x is said to have a derivative in the point x 0 if the forward or backward derivative exists. If a function has a derivative in a point the function is continuous in the point. But the converse is not true as will presently be shown. If the difference-quotient 1 becomes infinite as Δx approaches 0 the function is said to have an infinite derivative. In particular we may have lim Δx = 0 + f x 0 + Δx - f x 0 = + or - Δx and similarly for limδx = 0 -. When however we say of a function hat eine Ableitung verstehen sofern das Gegenteil nicht ausdrücklich bemerkt ist daβ that it has a derivative we shall use the word only in the sense of a proper derivative and exclude the case that the difference-quotient becomes eine endliche Ableitung d. h. ein eigentlicher infinite. Grenzwert vorliegt. If f x has a derivative at every point of an interval open or not the function is said to be differentiable in the interval. Sei die Funktion y = f x für alle Werte von x in einem Intervalle eindeutig erkl rt und seien x 0 x 0 + Δx zwei Punkte des Intervalls. Man bilde den Differenzen-quotienten Δy Δx = f x 0 + Δx - f x 0 Δx Wird f x 0 + Δx - f x 0 lim = f' x Δx = 0 Δ 0. x lim Δx = 0 f x 0 + Δx - f x 0 = + - Δ x so sagt man f x hat im Punkte x 0 eine unendliche Ableitung und nennt zum Gegensatz die eigentliche Ableitung eine endliche Ableitung. Wir werden jedoch unter den Worten f x Δx Δx 0 3 w = z a w = z m 4

11 w = z α w = z m w = z a 2 51 ~ 52 Consider the map defined by the function 1 w = z a whereis α is a positive real number. Let z = r cosφ + isinφ w = R cosφ + isinφ. Then 1 5 w = z m ~ 260 Ist m eine positive ganze Zahl so ist die Funktion f z = z m in der ganzen z-ebene eindeutig und analytisch. Da die Ableitung f' z = mz m - 1 im Punkte z = 0 sonst aber nirgends verschwindet sofern m > 1 ist so erweist sich damit die Abbildung der Umgebung eines beliebigen Punktes z 0 0 auf die w-ebene als ein-eindeutig und kon- R cosφ + isinφ = r a cosαφ + isinαφ. form. Hence 2 R = r α Φ = αφ + 2kπ. Setzt man z = r cosφ + isinφ Thus a circle about the origin z = 0 goes over into a circle about the origin w = 0. Consider a sector of a circle w = R cosφ + isinφ So führt die Gleichung w = z m 3 0 r r 1 0 φ φ 1 zu den Relationen Let = αφ and assume that αφ 1 = π 1 < α. Let r 1 = 1 then R 1 = 1. Thus a sector of the unit circle in the z-plane whose angle is φ 1 1 R = r = R 1 rm m { Φ = mφ φ = Φ + 2kπ { = π /α is m open out like a fan on a semicircle. And yet not wholly like a fan for the points of the z-figure are wo k = 0 1 m-1 ist. Wir wollen hier Φ zun chst auf das Intervall drawn in toward the centre. If for example 0 Φ π α = 2 the points on the circle r = 1 beschr nken und zugleich k = 0 nehmen. Dadurch wird φ zu einer eindeutigen Funktion von Φ in diesem Intervall. Bei dieser Festsetzung go over into 2 points on the circle R = 1 wird eine beliebige innerhalb des Winkels 0 φ π / m gelegene Figur 4 der z-ebene ein-eindeutig und konform auf eine Figur der oberen H lfte der w-ebene abgebildet. Insbesondere heben wir zwei Bereiche hervor a der Sektor des Einheitskreises 0 r 1 0 φ π /m wird auf den Halbkreis 0 R 1 0 Φ π bezogen b dem Winkel 0 φ π / m entspricht die ganze Halbebene 0 Φ π. 4 a m w = z a w = z m w = z a w = z m 1 5 w = z a R r Φ φ R r Φ φ

12 ε δ Institute for Advanced Study Marie M. Johnson National Mathematical Magazine Chelsea Publishing Company Ⅰ 1937 ~ 1938 Ⅱ 1937 ~ ~ ~ ~

13 Osgood W F. Functions of Real Variables M. Beijing University Press The National University of Peking Osgood W F. Functions of a Complex Variable M. Beijing University Press The National University of Peking Some Aspects of the Mathematical Exchanges between China and the United States in Modern Times A.. M ~ M ~ M ~ A.. 20 M ~ Walsh J L. William Fogg Osgood A. Peter Duren ed. A Century of Mathematics in America C. Part Ⅱ. Providence Rhode Island American Mathematical Society A.. C Koopman B O. William Fogg Osgood In Memoriam J. Bulletin of the American Mathematical Society ~ A. R. BD The Agreement A. R. 1 Abel 8 16

14 14 35 BD Letter from T. H. Kiang to G. Birkhof on August R. Cambridge Harvard University Archives HUG R... 1 ~ A. R. BD Z A.. Z ~ A. R. BD A. R. BD N A. R. BD N A R. BD Z ~ N Osgood W F. Introduction to the Calculus M. New York The Macmillan Company Osgood W F. Advanced Calculus M. New York The Macmillan Company A R. BD Osgood W F. Lehrbuch der Funktionentheorie M. erster band. Leipzig und Berlin Verlag und Druck von B. G. Teubner J Johnson M M. Functions of Real Variables by William Fogg Osgood J. National Mathematics Magazine ~ Osgood W F. Functions of Real and Complex Variables M. New York Chelsea Publishing Company A.. C

15 1 15 William Fogg Osgood and the Dissemination of Theories of Functions in China GUO Jinhai Institute for the History of Natural Sciences CAS Beijing China Abstract As a research professor of the National University of Peking William Fogg Osgood 1934 ~ 1936 the well-known American mathematician and Harvard University mathematics department professor taught at the mathematics department of the National University of Peking from 1934 to He mainly offered courses on theories of functions and was the first foreign mathematician to systematically disseminate the theories of functions in China. During this period he wrote two books published by the University Press in 1936 Functions of Real Variables and Functions of a Complex Variable. Some of the content of these two books originated from his Lehrbuch der Funktionentheorie though considerably adapted. They are the earliest textbooks on these subjects published in China. The manuscript of Functions of Real Variables was his teaching materials for the course on theories of functions of real variables. Functions of a Complex Variable is closely related to the content of the course of theories of functions of a complex variable. His teaching activities not only made the department's courses about theories of functions more specialized promoted the dissemination of theories of functions in China but also gave students training in the mode of Harvard University narrowing the gap between them at the international level. Key words William Fogg Osgood mathematics department of Peking University Functions of Real Variables Functions of a Complex Variable dissemination of mathematics

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