I am trying to use complex contour integration to calculate $\int_\mathbb{R} \frac{1}{1+t^4}e^{-itx}dt$.
I have done the full calculation for $x>0$ and ended up with a function that is not necessarily real, namely $-\pi i/2(e^{-7\pi i/4} e^{-i e^{5 \pi ix/4}}+e^{-5\pi i/4} e^{-i e^{7 \pi ix/4}})$.
I computed the residues in the lower half plane (2 of the 4 of them) and ended up with this, with the negative sign in front accounting for the reversal of the orientation of the contour.
To compute those residues, one at $e^{5 \pi ix/4}$ and one at $e^{7 \pi ix/4}$ I factor the $1+t^4$ and the rest is algebra. I obtain as the residues:
at $e^{5 \pi ix/4}$:
$e^{-7\pi i/4} e^{-i e^{5 \pi ix/4}}/4$
and for $e^{7 \pi ix/4}$:
$e^{-5\pi i/4} e^{-i e^{7 \pi ix/4}}/4$
The trouble I see is that the integral should be real, but as long as there are two competing residues in either the upper half plane ($x<0$) or lower half plane ($x>0$) with different real parts, then the imaginary part of the exponential with x in it will be different, and thus it cannot be real for all $x>0$. What am I missing?
Call the zeros of $1+z^4$ in the lower half-plane
$$\zeta = e^{5\pi i/4} = -\frac{1+i}{\sqrt{2}},\text{ and } \eta = e^{7\pi i/4} = \frac{1-i}{\sqrt{2}} = i\zeta.$$
Since the zeros are simple, the residues of $\frac{e^{-ixz}}{1+z^4}$ in these points are
$$\frac{e^{-ix\zeta}}{4\zeta^3} = -\frac{1}{4}\zeta e^{-ix\zeta} = \frac{1+i}{4\sqrt{2}}\exp\left(\frac{-1+i}{\sqrt{2}}x\right) = \frac{e^{-x/\sqrt{2}}}{4\sqrt{2}} (1+i)\left(\cos \frac{x}{\sqrt{2}} + i\sin \frac{x}{\sqrt{2}}\right)$$
and
$$\frac{e^{-ix\eta}}{4\eta^3} = -\frac{1}{4}\eta e^{-ix\eta} = \frac{-1+i}{4\sqrt{2}}\exp \left(\frac{-1-i}{\sqrt{2}}x\right) = \frac{e^{-x/\sqrt{2}}}{4\sqrt{2}} (-1+i)\left(\cos \frac{x}{\sqrt{2}} - i\sin \frac{x}{\sqrt{2}}\right)$$
respecitvely.
In that form, it is not hard to see that the real parts of the two residues cancel, and therefore the integral is real-valued for all $x > 0$. The case for $x < 0$ is analogous or can be obtained without computation by conjugation.