Abstract
In the proceeding chapter, we have discussed some special types of bilinear transformations. As noted earlier that these transformations are the most powerful tools for transforming circular regions in the z-plane into circular regions or half-planes in the w-plane. In this chapter, we deal with more general situations in which we shall answer more abstract questions for determining whether and in what manner a given finite portion of an analytic surface could be represented on a portion of a plane (This well-posed problem was treated for the first time by Riemann (1826–1856) in his integral dissertation of Göttingen in 1851, which is indeed a decisive turning point in the history of Conformal Representation.).
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils
N. H. Abel (1802–1829)
All the measurements in the world are not equivalent of a single theorem that produces a significant advance in the one greatest of sciences
Karl Friedrich Gauss (17177–1855)
(Generally ranked with Archimides and Newton at the pinnacle of mathematical achievement)
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Notes
- 1.
This well-posed problem was treated for the first time by Riemann (1826–1856) in his integral dissertation of Göttingen in 1851, which is indeed a decisive turning point in the history of Conformal Representation.
- 2.
Gesammelte Abhandlungen (Berlin, 1890), II.
- 3.
See, for example, Conformal Representation by C. Caratheodory, Cambridge Univ. Press, 1952.
- 4.
It was first employed to study flows around airplane wings by the pioneering Russian aero-and hydrodynamics researcher Nikolai Zhukovskii (Joukowski).
- 5.
Arthur Cayley (1821–1895), English mathematician and professor at Cambridge, is known for his important work in algebra, matrix theory, and differential equations.
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Pathak, H.K. (2019). Conformal Mappings and Applications. In: Complex Analysis and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-13-9734-9_7
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