Diffraction of Electromagnetic Waves By a Circular Aperture in An Infinitely Con

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Diffraction of Electromagnetic Waves By a Circular Aperture in An Infinitely Con
Jack Bazer
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To exemplify our procedure, we show how the behavior of V^ is obtained. We begin by assuming that | z/a | < e, where e < r/a is a small positive number. Then we deform the contour near t = in the right member of equation (3-1) into a semi-circle of radius e centered at t = and falling below the real t-axis. On the new contour -- C say — the ratio | z/a | will then be everywhere less than |t|. Then, we observe that the integral representation of V.^ [see equation (3. 1)^ niay ^^ expressed as (3. 16) V. (m) -19- ir\ j expiik R (r, z ;t) ] ^^ R (r, z;t) e 1 dt Vz+iat m 4"^(t)dt, on noting that (3-17) 1 S_ f expji'kR'^) r Sr V p+ ia(z+iat) dt d / eyp(ikR R + integrating by parts m times and finally making use of the "behavior of f|"^''(t) at t = ±1 [see equation (3. 3), and recall that f|™^(t) is oddj . Next, we make use of the fact that exp(ikR )/r = (z+iat) d(exp ikE )/dt, integrate by parts once again and find that , +/ f3 18) V^™^ = - — l—]'^ r expfikR""(r, z;t)] ^^' ' 1 ka Va / I z+iat dt \ z+iat/ m f^"^)(t) -1 expfikE (r, z;t)] R"^(r, z;t) dt V z+iat -im+l f{"^)(t)dt.

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