Download Plane Elastic Systems by Louis M. Milne-Thomson PDF

By Louis M. Milne-Thomson

In an epoch-making paper entitled "On an approximate answer for the bending of a beam of oblong cross-section less than any approach of load with designated connection with issues of centred or discontinuous loading", bought by means of the Royal Society on June 12, 1902, L. N. G. FlLON brought the suggestion of what was once consequently known as by means of LovE "general­ ized aircraft stress". within the related paper FlLO~ additionally gave the elemental equations which exhibit the displacement (u, v) when it comes to the complicated variable. the 3 uncomplicated equations of the speculation of KoLOsov (1909) which was once for this reason constructed and greater by means of MUSKHELISHVILI (1915 and onwards) could be derived without delay from Filon's equations. The derivation is indicated via FlLO)!E~KO-BoRODICH. even if FILO)! proceeded right away to the true variable, traditionally he's the founding father of the fashionable idea of the applying of the advanced variable to airplane elastic difficulties. the strategy was once constructed independently through A. C. STEVEXSOX in a paper bought through the Royal Society in 1940 yet which was once now not released, for defense purposes, till 1945.

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12. Cauchy's integral fonnula 55 This theorem remains true if from the interior L of C we omit one point z (or a finite number of such points), provided that lim (t - z) j(t) = (2) 0, t->-z when t tends to z by any path lying entirely inside C. For given a positive number s we can choose r = It - zl so small that Ij(t) I < sir, and so the modulus of the integral ofj(t) taken round the circle It-zl = r is less than 2ns. 12 C. The function j(t) satisfies (2) and is holomorphic everywhere within C except at z, where it is undefined.

Cauchy's integral fonnula 55 This theorem remains true if from the interior L of C we omit one point z (or a finite number of such points), provided that lim (t - z) j(t) = (2) 0, t->-z when t tends to z by any path lying entirely inside C. For given a positive number s we can choose r = It - zl so small that Ij(t) I < sir, and so the modulus of the integral ofj(t) taken round the circle It-zl = r is less than 2ns. 12 C. The function j(t) satisfies (2) and is holomorphic everywhere within C except at z, where it is undefined.

More generally we shall designate by L the region on the left of an observer who describes C in a prescribed sense. 11. The complex Stokes's theorem If the closed contour C is replaced by an open arc B D described in the sense B to D, the region L can often be conveniently defined by considering the arc as prolonged indefinitely by drawing tangents to it at Band D fig. 10 (ii), or by joining B to D by an arc to form a closed curve figs. 10 (iii), (iv). / 'D / / . _- ----~ ß Fig. 10 (ii) Fig. 10 (iii) Fig.

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