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By E. F. (Ed.) Beckenbach

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Extra resources for Construction and Applications of Conformal Maps, Proceedings of a Symposium

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1 4 . 50) T h e length e of most core legs is relatively large as compared with the width, so that the field is ractically uniform for y = — a . 37). 52) y /I / J * -1 2(r/X) cos 5 . 54) T h e accuracy of this formula shall be checked with the corresponding results obtained for two whole upper frames (fig. 14) one of which has a finite opening width rf, but an infinite yoke height, c= «>, while the opening of the other frame is reduced to a sht, rf=0, whereby c remains finite. T h e frame with c = « represents a fork-shaped space with two equal channels, c=:a, and potentials |3=0, 7 = — a .

2. Some General Rules of Field Mapping by the Use of Complex Functions as Solutions of the Field Equations The proper mathematical method for the mapping of these fields can be selected on the basis of their general ec|uations. Consider first the electric field between the windings (fig. 2). This field is fully described b y the vector of the electric field strength and flux density D. 1) D=kE. This field is excited b y electric charges which are located on the conducting surfaces and which are sources in the field.

29) Xjt_ P*-i T h e quantities a, . X, . 26). 26), we find that y = const, i n a simply connected region. const, in a doubly connected region, y=C\/i-t-Cy2-|-const, i n a triply connected region, 80 that the field is of zero strength i n a simply connected region, has only one pattern if the space is doubly connected, but may assume an infinite number of patterns i n a triply or more highly connected domain. 3. Simplications in the Mapping of Rectangular Frames This group of field problems includes the mapping of doubly and triply connected rectangidar frames (fig.

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Construction and Applications of Conformal Maps, Proceedings of a Symposium by E. F. (Ed.) Beckenbach


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