The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was 80% and 117%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible.
The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows.
There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range.
One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper Summa de arithmetica, geometria, proportioni et proportionalita (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. ).
We can see, that already then, they were definitely important (complex numbers).
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 Miś B. (2008): Secret e number and other secrets of mathematics.– Scientific and Technical Publishing House, Warsaw.
 Thompson P., Papadopoulou G. and Vassiliou E. (2007): The origins of entasis: illusion, aesthetics or engineering? –Spatial Vision, vol.20, No.6, pp.531-543.
 Marcinowski J. and Sadowski M. (2016): Buckling capacity of non-prismatic rods with polygonal cross-sections.– In: Sustainable Construction, the University Publishing House of the University of Technology and Life Sciences in Bydgoszcz, Bydgoszcz.
 Krzyś W. (1968): Optimale formen gedrückter dünnwandiger stützen in elastisch-plastischen bereich. – Wiss. Z. Tech., Univ. Dresden, vol.17, No.2, pp.407-410.
 Gajewski A. and Życzkowski M. (1998): Optimal structural design under stability constraints.– Kluwer Academic Publishers, Dordrecht, Boston, London.
 Bochenek B. and Krużelecki J. (2007): Optimization of construction stability. Contemporary problems.– Cracow University of Technology Publisher, Cracow.
 Marcinowski J. and Sadowski M. (2015): Shape optimization of non-prismatic rods of circular hollow cross-sections and of variable wall thickness.–In: Proceedings of the stability of structures: XV-th symposium. Zakopane, Poland, 2018. Łódź: Department of Strength of Materials and Structures of the Lodz University of Technology, pp.99-100.
 Gliński H., Grzymkowski R., Kapusta A. and Słota D. (2012): Mathematica 8. – Publishing House of the Jacek Skalmierski Computer Laboratory, Gliwice.
 Opara K. (2014): Analysis of the differential evolution algorithm and its application in the determination of statistical dependencies.– Abstract of the Doctoral Thesis.
 Bobrowski C. (1995): Physics – Short Course.– Warsaw: Scientific and Technical Publishing House.
 Stasiak J. and Walden H. (1971): Mechanics of liquids and gases in sanitary engineering. – Arkady Publishing House, Warsaw.
 Rykaluk K. (2012): Problems of stability of metal structures. – Lower Silesian Educational Publishing House, Wrocław.