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

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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. 6. )-1 (Z - t). 6) and the series for p* takes the form which may be a final one: z p* (A, 0) = go (A, 0) - Re if>o (t) [H (iX, {3, y; x)] dt}. U o ! 1. 7) can, after some elementary transformations, be written in the form: p* = Re {if>o (Z) - if>o (Z) H + (iX, {3, y; 0) z + if>o (0) H [iX, {3, y; (2 ).

4) there is an arbitrary function of one complex variable and in equation (3. 5) two arbitrary functions of one real variable. The former remark follows, since in order to obtain real solutions, g must be chosen the conjugate to f; that is g = f (ft - i D). 1. 3) can be reduced to the form Po 0 + lJIA, A, = 0 and the general solution is lJI = f [i (1 - fl)'/2 - D] + g [i (1 - fl)"o + D]. 1. 6) 31 3. Solution of the Differential Equation 2. 4) the variables log q and () instead of q and 0, i. , P q q-1 p]Ogq and Wq = q-1 W]Ogq.

6) 45 4. ~:'). ~n) are given in 6, p. 269. )'/3 for k = 1·4. This choice yields T = e If these series are substituted in (1. 111 = 1. + + . t 4. 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.

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