# Bergman’s Linear Integral Operator Method in the Theory of by M. Z. v. Krzywoblocki Sc. D. (Lille), Ph. D. (Brooklyn), M.

By M. Z. v. Krzywoblocki Sc. D. (Lille), Ph. D. (Brooklyn), M. A. (Math., Stanford), M. S. (Appl. Math., Brown) M. Aer. En. (Brooklyn), Dipl. Ing. (Lemberg) (auth.)

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Term-by-term) derivatives converge uniformly. , the function A (i) is a dominant of A (i), which fact is symbolized by writing A ~ A, i E I. ~. d. n [d. 4) IE(1) (i)1 ~ EO) (i), i and also II Thus E(l) (i) is a dominant of E 1<- E(l) (i), i. 1) ~ E(l). 1b) O~H(n)(i). 6) ~E(n+1)(i), dE(n + 1) I::s;: dJjj(n + 1) • d. d. 6a) + ]iiI jfj(n)] E(n + 1) (i) ~ E(n+ 1) (i). 7) Thus starting with the assumption that E(n)~ E(n), we have proved by induction that E(n + 1) ~ iff(n + 1). 2), i. 9) ·00 where the cn's are some conveniently chosen positive constants, which will be determined later.

The computation may now be relatively easy, while we may have to justify our choice of by showing that the hypothetical gas thus described is sufficiently similar to a real physical gas. 1. 1) E. T. Copson: Functions of a complex variable. Oxford: The Clarendon Press. 1935. 24 II. Simplified Pressure-Density Relation t instead of the function as defined and calculated in previous sections. 1. 2) Go = 1, Gn 00) (- = 0, n > 0, whence the general formula: Gn = n! fl" (- A)-n. 3) Using the function U (1.

1a) IToo-TI < , which is bounded by two straight lines: ; 7: - 0 - 7:00 = 0, ~ 7: + 0 - 7:00 = O. The intersection of both domains determines the domain in which both series converge. The upper limit on the positive part of the (2 A)-axis is given by the quantity 7:0 = 2 Ao, (i. , vertical line). In a particular case we may assume 7:00 = 0, which implies that the domain of convergence is a triangle bounded by the following three ! lines: 7: ± 0 = 0, and 7: = 7:0, The last question which should be mentioned is the investigation of the behavior of the function Fl (2 A).