Abstract
By a domain we mean a connected open subset of the complex plane. The correspondence (x, y) ↔ z = x + iy between the real plane \(\mathbb {R}^2\) and the complex plane \(\mathbb {C}\) allows one to regard a complex-valued function of a complex variable as
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a map from a domain \(D \subset \mathbb {C}\) in the complex plane to the complex plane \(\mathbb {C}\) (notation: w = f(z));
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a map from a domain \(D \subset \mathbb {R}^2\) in the real plane to the complex plane \(\mathbb {C}\) (notation: w = f(x, y));
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a map from a domain \(D \subset \mathbb {R}^2\) in the real plane to the real plane \(\mathbb {R}^2\) (notation: (u, v) = f(x, y), u = u(x, y), v = v(x, y)).
In what follows, we will often switch between these interpretations.
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Natanzon, S.M. (2019). Holomorphic Functions. In: Complex Analysis, Riemann Surfaces and Integrable Systems. Moscow Lectures, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-34640-9_1
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DOI: https://doi.org/10.1007/978-3-030-34640-9_1
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