In this paper, we first define the Teodorescu operator related to the Helmholtz equation and discuss its properties in quaternion analysis. Then we propose the Riemann boundary value problem related to the Helmholtz equation. Finally we give the integral representation of the boundary value problem by using the previously defined operator.

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In this paper, we define the two‐sided fractional Clifford–Fourier transform (FrCFT). Using its properties, we get some uncertainty principles of the FrCFT. Two parts are obtained. One part is a modified uncertainty principle. The uncertainty principle states a lower bound on the spreads of two specific transform domains. It is shown that only a Gaussian‐type signal minimizes the uncertainty. We also give a Heisenberg‐type uncertainty principle. The other part is a logarithmic uncertainty principle, which may be obtained from a sharp of Pitt's inequality.