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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**Extra resources for Bergman’s Linear Integral Operator Method in the Theory of Compressible Fluid Flow**

**Example text**

9) This means that the function F2 (i*) has a singularity for A = 0 or i* = 2 A* = = - (h -1 - 1) n, corresponding to an extreme case M --'> 00. This transformation may be very convenient in some cases and a more thorough discussion of it may be found in 1. Let us mention that because the quantity A = 1/2 (h -1 - 1) n represents an upper bound for A, it is obvious that for the upper bound of the variable T, the quantity io = (h- 1 - 1) n may be chosen. That quantity ro should be used in the considerations presented in section III.

2) converges very slowly, and it is therefore necessary to employ a large number of terms in order to obtain a good approximation for T*. 7), the number m must be chosen rather large. If this is the case, it is then expedient to replace the expansion (1. 2) by (1. 9). Theoretically, this is, however, not the only way of overcoming this difficulty, and in the following other means of doing so will be indicated: this alternative approach employs the method of analytic continuation. Let T (q, e) be determined in a domain, say H, and let Tn> n = 1,2, ...

Thus for A = 0, the quantity B = 0 and the function Fl -- 00. In some cases, however, it is possible to overcome this difficulty by shifting the origin. 1) '/ - '/ 2' so that A* will mean A* = A - ;, 7:* = 7: - a. 2) W 3*'1* + F2 (3*, 'YJ*) W = 0, 1 1 where W (3*, 'YJ*) = p* (3* + '2 a, 'YJ* + '2 a), F2 (7:*) = Fl (7:* + a), so that F2 is analytic for 7:* = O. * .. * E(n+l) (7:*) .. 3a) 1) (7:). 39 10. Transformation to the Physical Plane The domain of convergence, remaining the same, referred to the r* system, is merely a shift of the original domain.