'Show that the heat equation has complex-valued solutions of the form

F(x) exp(iwt) provided that kF′′ = iwF. Find F if

F(x) tends to 0 as x tends to 1 and F(0) = A, where A is a real constant.'

Are you sure about the highlighted part? If "F(x) tends to 0 as x tends to infinity" then the given solution can be obtained.
I can show the first part of this, but i'm having issues with the next bit, what I get must be wrong, because I can't seem to split it into real and imaginary for the second part of the question

'Let T(x, t) be the real part of F(x) exp(iwt). Verify that:

Any help would be greatly appreciated!