Double Integral Over The Whole Plane

Calculus Level 5

\[ \large \dfrac{1}{2 \pi \hbar}\iint\limits_{\mathbb{R}^2} \left (q-\frac{x}{2} \right )ke^{\frac{i}{\hbar}(p-k)x} \, dx dk\]

Let be \(p\), \(q\) are real parameters; \(\hbar\) denotes the Planck's constant, but for this exercise, you can treat it as another (real) parameter. Evaluate the integral above.

Clarification: \(\displaystyle \iint\limits_{\mathbb{R}^2}\) simply means \(\displaystyle \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\).

Hint: Fourier transform.

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