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^^ yeah he has it.
In your Y*, the complex number change to negative, when you do Y*xY your Im part goes out.
YxY* = (...)^2x(..)^2 = [(b/Pi)^1/2] x [e^-bx^2]
Primitive of [e^-bx^2] is [(Pi/b)^1/2], so in the end you have 1