1162 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



In the present case this leads to 



dH, = -(47r/^/X) J %-T-T^ T^ cos (27rx/X) dx dz 



(xo - xY + (zo - zY 



(8) 



dH, = -(4irIm/\) -, ^^. 7 f r2 COS {lirx/X) dx dz 



{xq - xY + (zo - z)^ 



The total field at {xq , Zo) is obtained by integrating with respect to x 

 over the range — <» to + oo and with respect to z over the range —5/2 

 to +5/2. In carrying out the integration over x it is convenient to make 

 the substitution 



(xo - x)/{zo - z) = p 



(9) 

 dx = — (zo — z) dp 



Neglecting terms which obviously integrate to zero, this gives 

 H, = (4x/„A) sin (2xVX) f' [ O sin [M., - .) j>Al I ^^ 



S. = {4xWX) COS (2WX) /■"' r r '"" t^"^^-/^^/^J dJrf. 



J-«/2 Uoo 1 + ^^ J 



20 > 2 



The integrals in brackets can be found in tables. ^° Carrying out the inte- 

 gration gives 



a/2 



-a/2 

 .a/2 



H, = -{4tIJ\) sin (27r:x^/X) f 



£r. = -(47rVx) cos (27rVX) / e-''^''-'^'^ 

 Zo > z 



(11) 



dz 



J-S/2 



which integrate to 



H. = -2^/„ sin (27rVX)^"'"°^'[e'*^' - ^"''^1 



£r, = -2x7^ cos (27rVX)«"'"°^'[«'*^' - ^"'*^'l 



2o > 5/2 



(12) 



'» D. Bierens de Haan, "Nouvelles Tables D'Integrales D^finies," p. 223, Leide, En- 

 gels, 1867. 



