Tight Binding Description of Graphene (Hamiltonian and Overlap Matrix)

[nyxynyx]

http://edu.ioffe.ru/register/?doc=galperin/l3pdf2.tex

I don't understand how do you get from equation 21 to 22? How did the summations of exponentials becomes cosine functins?

 

[DrDu]

Using the values of the I, the summations can be seen to be geometric series. If you do them explicitly, you will recover the cosines.

 

[asheg]

exponent of imaginary is summation of sine and cosine of real

exp(ix)+exp(-ix) = 2cos(x)