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By Robert L. Taylor

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Example text

Then oj/oi = 0 and fj(Z)dZ=O. (7) This is Cauchy's theorem. Corollary (ii). Let (7) hold for every contour C within a region D. Then for an arbitrary area S within this region Therefore oj/oi = 0 and the functionjis a holomorphic function of z inR. This is Morera's theorem. Although the demonstrations in these corollaries are of a frankly heuristic character, they are nevertheless enlightening. 12. Cauchy's integral formula Consider a function j(t) of the complex variable t which is holomorphic inside and upon the simple c10sed contour C.

In particular for an infinite plate with one hole rotational dislocation is impossible. Apply (7) to (5) and (6) and we see further that the expressions Sand Twhere S = h'(z) + 1i'(i) + 4 (2- v) ::~2 T = ih"(z) + k"~Z)-4'v ~:? ' I (9) must be bounded at infinity. Let us examine the implication of this in the case of gravity and steady rotation. 11 we have 40Z Q ozo2 1 =-T n 2 e ( z-a) (- -) 1 . 1 i- z-a +Tgee-""z+Tge elXz and if S is to be bounded at infinity, it is evident that we must have n=O, (10) since no term in S can cancel the term n2e zZ ..

51 Examples II 4. The boundary of an elastic body is acted on by normal pressure p and shear t at the element ds. Obtain the boundary condition in the form :5 {'W(z) +zW(z)+ 'w(z) - 8 ~~-/) ~;} = -2 (P + it + V) :; and find the condition at an unstressed boundary in the absence of a body-vector field. 5. Determine the cyclic functions of zlnz, zlnz, zlnz, zlnz+Zlnz with respect to a circuit C which encloses the origin. 6. Elastic material is in the form of an annulus. Discuss the terms in the complex stresses W (z), w (z) which will lead to (i) a translational dislocation t (ii) a rotational dislocation c, and illustrate by a diagram.

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