THE CALCULATION OF MODULATION PRODUCTS 



231 



over which the integration is performed. The non-zero values of 

 /(x, 3^) lie in the shaded region of Fig. 1. From the symmetry of the 

 region about the x and y axes we deduce at once that the sine coeffi- 

 cients, B±,nn, must vanish and that the cosine coefficients, A±rnn, may 

 be obtained i)y integrating throughout one quadrant only and multi- 



Y 



Pig 1— Region of integration for the determination of the coefificients in the double 

 Fourier series expansion of/ (-v, y). 



plying by four. We therefore obtain, on substitution of the proper 

 limits, 



2P r" 

 Amn = A±„,n = -^ \ COS njdy 

 ^ Jo 



J '•arc cos (— fr cos v^ 

 (cos X -\- k COS y) COS mxdx (9) 

 



