Identities between field-equations in the general field theory of Schouten and van Dantzig and certain results in the theories of legendre and confluent hypergeometric functions

The general field- theory of Schouten and van Dantzig forms an important step in the solution of the unification problem of the gravitational and electromagnetic phenomena in Physics. This theory depends on the use of a projective geometry employing five homogeneous coordinates. In the first part of this thesis an attempt is made to make a contribution to this unified field- theory by finding the identical relations between the field- equations in this theory. We have also shown the connection between these identities and the identities found by Professor E.T. Whittaker between the field- equations of Einstein's general relativity.In the second part of the thesis we first develop certain series and integral properties of Legendre functions in a direction, which has received little attention till now. The import - á.nce of the properties of Rₘ -functions developed towards the end of the second part may be seen from the following remark of Professor E.T. Whittaker in his well -known paper (The Bulletin of the American Yathematical Society, volume 10, [1903 -04,] page 133) in which he defines the function Wᵣ,ₘ (3):"There are other members of the family of functions Wᵣ,ₘ (3) which have not been noticed, but which give promise of interesting properties. Among these may be mentioned the families of functions for which rn, = 0 and those for which m = 1/2 ".The Rₘ -functions considered correspond to the case m = 1/2, the associated differential equation having arisen recently in the theory of turbulence in researches of W. Tollmien and Th. von Kármán and also in the wave- mechanical theory of the α -particles by Theodor Sexl

Microwave device investigations

Several tasks were active during this report period: (1) noise modulation in avalanche-diode devices; (2) schottky-barrier microwave devices; (3) intermodulation products in IMPATT diode amplifiers; (4) harmonic generation using Read-diode varactors; and (5) fabrication of GaAs Schottky-barrier IMPATT diodes

On uniformization of Burnside's curve y2=x5−xy^2=x^5-x

Main objects of uniformization of the curve $y^2=x^5-x$ are studied: its Burnside's parametrization, corresponding Schwarz's equation, and accessory parameters. As a result we obtain the first examples of solvable Fuchsian equations on torus and exhibit number-theoretic integer $q$-series for uniformizing functions, relevant modular forms, and analytic series for holomorphic Abelian integrals. A conjecture of Whittaker for hyperelliptic curves and its hypergeometric reducibility are discussed. We also consider the conversion between Burnside's and Whittaker's uniformizations.Comment: Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic functions has been moved to arXiv:0808.348

On some definite integrals involving Legendre functions

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