Fourier transform of a piecewise continuous function

Question

$$ f(x) = \begin{cases} 0, & x<3 \\ x, & 3\le x\le 5\\ 0, & x>5 \end{cases} $$

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$=\int^5_3 x e^{-1\xi x}$ $dx$

$=-\frac{x}{i \xi} + \int \frac{e^{-1\xi x}}{i \xi}$ $dx$

$=-\frac{x}{i \xi}+\frac{e^{-1\xi x}}{\xi^2}|^5_3$

$=e^{-5i\xi}(\frac{1}{\xi^2}-\frac{5}{i\xi})+e^{-3i\xi}(\frac{3}{i\xi}-\frac{1}{\xi^2})$

But the answer is

$=(5i\xi^{-1}+\xi^{-2})e^{-5i\xi}-(3i\xi^{-1}+\xi^{-2})e^{-3i\xi}$

Answer 1

You have the same solution but there is difference between your solution and the answer in terms of notation

• The imaginary unit is the numerator in the given answer
• The fractions are written as negative powers in the given answer