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This snapshot is from the book Solid State Physics by Ashcroft,Mermin. They represented the Field and Momentum as the real part of a complex function. As far as what I understand, we represent fields in complex notations for our own convenience over the standard real representation which results in computational complexities. Why then does the imaginary part of the field which we have designated by ourselves need to satisfy the equation 1.24 which the real part of the field(the actual field) satisfies?