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Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions. Specifically, the algorithm cannot solve the constant problem, which is undecidable when it needs to determine whether an arbitrary complex number expression is equal to zero. The complete description of the Risch algorithm takes over 100 pages. The Risch–Norman algorithm is a simpler, faster, but less powerful variant that was developed in 1976 by Arthur Norman. Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller.
Robert Henry Risch (born 1939) is an American mathematician who worked on computer algebra and is known for his work on symbolic integration, specifically the Risch algorithm. This result was cited as a milestone in the development of mathematics: Calculus students worldwide depend on the algorithm, whenever they appeal to Wolfram Alpha to do their homework. He is also known for results on algebraic properties of elementary functions. He received his PhD from the University of California, Berkeley in 1968 under the supervision of Maxwell A. Rosenlicht. After his PhD, he worked at the Thomas J. Watson Research Center Mathematics of AI group and, between 1970 and 1972, the Institute for Advanced Study.
In mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these. Some functions which are encountered by beginners are not elementary, such as piecewise-defined functions. More generally, in some modern treatments, elementary functions comprise the set of functions previously enumerated, all algebraic functions, and all functions obtained by roots of a polynomial whose coefficients are elementary. The elementary functions were originally defined by Joseph Liouville in 1833. A key property is that all elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules (or the rules for implicit differentiation in the case of roots). The Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined (possibly with multiple values, such as the elementary function z {\displaystyle {\sqrt {z}}} or log z {\displaystyle \log z} ) for every complex argument, except at isolated points. In contrast, antiderivatives of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative. Liouville's result is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. The Risch algorithm (1968) can decide whether an elementary function has an elementary antiderivative, and, if so, to compute it. However, as of 2025, there is no full implementation.
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Joe Louis.