The magister's thesis of Sokhotskii was the first research paper on complex analysis published in Russian. It contains many important results which were later ascribed to other mathematicians. First of all, there is the famous theorem on the behaviour of an analytic function in a neighbourhood of an essential singularity. This theorem was published by Sokhotskii (in his magister's thesis) and by Casorati in 1868, whereas Weierstrass published it eight years later - in 1876. Furthermore, Sokhotskii was the first to apply the calculus of residues to Legendre polynomials. The credit for this procedure is usually given to Hermann Laurent. Finally, the so-called Plemelj formulas are also due to Sokhotskii, who published them in his doctor's thesis in 1873, that is to say 35 years before Plemelj.Following the award of his Master's degree (essentially equivalent to a Ph.D.) he began teaching at the University of St Petersburg. The courses he gave in 1869-70 included the first course on the theory of functions of a complex variable to be taught in that university. He received a doctorate from St Petersburg in 1873. His doctoral dissertation On definite integrals and functions with applications to expansion of series was an early investigation of the theory of singular integral equations. It investigated in detail Cauchy-type integrals which played an important role in boundary value problems in the theory of functions of a complex variable. Youschkevitch writes :-
In his doctoral thesis Sokhotsky continued his studies of special functions, particularly of Jacobi polynomials and Lamé functions. One of the first to approach problems of the theory of singular integral equations, Sokhotsky in this work considered important boundary properties of integrals of the type of Cauchy and, essentially, arrived at the so-called formulas of I Plemelj (1908).Sokhotsky was appointed as an extraordinary professor after the award of his doctorate (essentially equivalent to the German habilitation) and then became an ordinary professor at the University of St Petersburg in 1883. From 1875 he also held a position at the Institute of Civil Engineers :-
His lectures, especially on higher algebra, the theory of numbers, and the theory of definite integrals, were extremely successful. Sokhotsky was elected vice-president of the Mathematical Society of St Petersburg in 1890 and succeeded V G Imshenetsky as president in 1892.Influenced by Chebyshev, he studied special functions, in particular Jacobi polynomials and Lamé functions. He also wrote a number of papers on the theory of elliptic functions and on theta functions. His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations. Other topics which Sokhotsky studied included Zolotarev's theory of divisibility of algebraic numbers in The application of the principle of the greatest divisor to the theory of divisibility of algebraic numbers (1898).
Article by: J J O'Connor and E F Robertson
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