Transactions of the American Institute of Electrical Engineers Volume 38, pt. 1

 
9781130003451: Transactions of the American Institute of Electrical Engineers Volume 38, pt. 1
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1919 Excerpt: ...problem discussed and solved on pages 405-409 of my paper. Now the problem involved in this operational expansion is two-fold; first to justify the expansion in the operator p itself, and secondly to determine (adopting Mr. Bush's notation) the significance of the symbols p"f. The first phase of the problem which can in no sense be neglected in a rigorous discussion, Mr. Bush ignores entirely and the second he begs by indentifying the symbol pn with the differential operator dn/dt" at the start of the discussion. This is all the more unjustified because he subsequently assigns non-integral values to n which makes the operator dn/dt" meaningless. Now it is true that if we start with the solution expressed as the Fourier integral given on page 359 of the paper it is possible to evaluate the symbolic expression p"f with some reservations involving the legitimacy of differentiating under the integral signs but this leaves the first phase of the problem still untouched. For we have merely shown that assuming the legitimacy of the expansion in the operator p, the series resulting is a formal solution. The real problem as I see it, still remains: to establish the legitimacy of the expansion in the operator p and then to investigate the significance of the resulting divergent series in inverse powers of t and its relation to the actual convergent solution. It is in the investigation of these important questions, without which the solution must remain in doubt, that I have found the application of integral equations, as developed in my paper, of the greatest value. Owing to the importance of the foregoing I shall elucidate it in connection with the specific problem stated and solved on pages 405-409 and employed by Mr. Bush in his discussion: nam...

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