The authors review some of the basic concepts and results of quaternionic analysis which can be found in a number of sources including the following books [\textit{K. Gurlebeck} and \textit{W. Sprossig}, Quaternionic analysis and elliptic boundary value problems. Mathematical Research, 56. Berlin: Akademie-Verlag (1989; Zbl 0699.35007)]; [\textit{V. V. Kravchenko} and \textit{M. V. Shapiro}, Integral representations for spatial models of mathematical physics. Pitman Research Notes in Mathematics Series. 351. Harlow: Longman (1996; Zbl 0872.35001)]; [\textit{K. Gurlebeck} and \textit{W. Sprossig}, Quaternionic and Clifford calculus for physicists and engineers. Mathematical Methods in Practice. Chichester: Wiley (1997; Zbl 0897.30023)]; [\textit{V. Kravchenko}, Applied quaternionic analysis. Research and Exposition in Mathematics. 28. Lemgo: Heldermann Verlag (2003; Zbl 1014.78003)]; [\textit{K. Gurlebeck, K. Habetha} and \textit{W. Sprossig}, Holomorphic functions in the plane and \(n\)-dimensional space. Transl. from the German. Basel: Birkhäuser (2008; Zbl 1132.30001)]. The authors illustrate applications of some of the presented facts considering certain boundary value problems for the Poisson equation as well as proposing a method for explicit evaluation of a class of integrals.