The New International Encyclopædia/Sturm, Jacques Charles François

STURM, stụrm, (1803-55). A French mathematician, born at Geneva. He was educated at the Academy of Geneva, and in 1827, with his friend Colladou, took the Grand prix de mathématiques for the best memoir on the compression of liquids. The famous theorem that bears his name was discovered in 1829. A statement of the results secured by this theorem requires the definition of Sturm's functions: If f(x) = 0 be freed from equal roots, and f(x) be divided by f ' (x) (the derivative of f(x)), and the last divisor by the last remainder, changing the sign of each remainder before dividing by it, until a remainder independent of x is obtained, or else a remainder which cannot change its sign, then f(x), f ' (x), and the successive remainders, constitute Sturm's functions. The theorem asserts that if, as x increases, f(x) passes through the value zero, Sturm's functions lose one change of sign; if any other of Sturm's functions vanishes, there is neither

loss nor gain in the number of changes of sign; the number of roots of f(x) = 0 between a and b is equal to the difference in the number of changes of sign in Sturm's functions, when x = a and when x = b. In 1838 Sturm began teaching in the Ecole Polytechnique, and two years later was elected to the chair made vacant by the death of Poisson.