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Invariance under a Reflection

Our candidates for eigenfunctions of the Laplacian are

cos(nX) cos(mY)   or   sin(nX) sin(mY) [Back].

To complete invariance under the entire group pmg, it suffices to achieve invariance under a horizontal mirror whose axis is a quarter of the way up the cell. The functions will have to satisfy:

f(x, L/2-y) = f(x,y).

Working out the trigonometry (just apply the angle addition identity), we find that we require m to be odd in the sine terms, with m even in the cosine terms.

[Next] Recipe for Invariance under pmg
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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 12 Aug 1998 --- Last modified: Sep 30, 2003 9:26:43 AM
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