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I would need the Fourier transform of the 'truncated Hann function' / 'Hann window'.

With 'truncated Hann function' / 'Hann window' I mean the Hann function h(t) with parameter 'a' from http://mathworld.wolfram.com/HanningFunction.html, but truncated so that it is zero outside of the interval [-a, a].

This 'Hann window' is a classical window function.

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    Assuming the Hanning function is given by $f(t) = \cos^2\frac{2\pi t}{2a}$, you should try substituting in a complex exponential: $$\cos\frac{2\pi t}{2a} = \frac{1}{2} (e^{i\pi t/(2a)}+e^{-i\pi t/(2a)}),$$ and computing. – 2017-01-25
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    Typo: should be $\cos\frac{\pi t}{2a}$ and $\cos^2\frac{\pi t}{2a}$ – 2017-01-25

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The continuous Hann function is

$$ f(x) = \begin{cases} \frac{1}{2} \left(1 + \cos (\pi x / a) \right) & -a \le x \le a \\ 0 & \text{otherwise} \end{cases} $$

where $a > 0$. This can be considered a product of two functions $f(x) = g(x) h(x)$ where $g(x) = \frac{1}{2} (1 + \cos (\pi x / a))$ and $h(x) = \operatorname{rect}(x / 2a)$, with

$$ \operatorname{rect}(x) = \begin{cases} 1 & |x| \le \frac{1}{2} \\ 0 & \text{otherwise.} \end{cases} $$

Using a table of transforms for common functions, the Fourier transform of $g(x)$ is a collection of impulses

$$ G(\xi) = \tfrac{1}{2} \left( \delta(\xi) + \tfrac{1}{2} \left[\delta(\xi-\tfrac{1}{2 a}) + \delta(\xi+\tfrac{1}{2 a})\right] \right) $$

and the Fourier transform of $h(x)$ is a sinc function

$$ H(\xi) = 2 a \operatorname{sinc}(2 a \xi) $$

where $\operatorname{sinc}(x) = \sin(\pi x) / \pi x$.

Since $\mathcal{F}(g h) = G * H$, and convolution with an impulse train produces a sum of shifted functions, this gives

$$ F(\xi) = a \operatorname{sinc}(2 a \xi) + \tfrac{1}{2} a \operatorname{sinc}(2 a \xi - 1) + \tfrac{1}{2} a \operatorname{sinc}(2 a \xi + 1) $$

This can be verified using Wolfram Alpha. Note that they use a different definition of the sinc function.