By Steven G. Krantz

This is a booklet approximately advanced variables that offers the reader a brief and available creation to the most important themes. whereas the assurance isn't finished, it definitely provides the reader a high-quality grounding during this primary sector. there are various figures and examples to demonstrate the critical rules, and the exposition is vigorous and welcoming. An undergraduate eager to have a primary examine this topic or a graduate pupil getting ready for the qualifying tests, will locate this e-book to be an invaluable source.

In addition to special rules from the Cauchy conception, the ebook additionally comprise sthe Riemann mapping theorem, harmonic capabilities, the argument precept, normal conformal mapping and dozens of different important topics.

Readers will locate this ebook to be an invaluable spouse to extra exhaustive texts within the box. it's a beneficial source for mathematicians and non-mathematicians alike.

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**Extra info for A Guide to Complex Variables**

**Sample text**

4. Of course one derives this more general version of Cauchy’s formula with the standard device of deformation of curves. A topological notion that is special to complex analysis is simple connectivity. We say that a domain U ⊆ C is simply connected if any closed curve in U can be continuously deformed to a point. Simple connectivity is a mathematically rigorous condition that corresponds to the intuitive notion that the region U has no holes. If U is simply connected, and γ is a closed curve in U, then it follows that γ can be continuously deformed to lie inside a disc in U.

1) Let U ⊆ C be a connected open set. Let f, g be holomorphic on U. If {z ∈ U : f(z) = g(z)} has an accumulation point in U, then f ≡ g. 2) Let U ⊆ C be a connected open set and let f, g be holomorphic on U. If f · g ≡ 0 on U, then either f ≡ 0 on U or g ≡ 0 on U. 3) Let U ⊆ C be connected and open and let f be holomorphic on U. 1) for every j ∈ {0, 1, 2, . . }, then f ≡ 0. 4) If f and g are entire holomorphic functions and if f(x) = g(x) for all x ∈ R ⊆ C, then f ≡ g. 4). 1) because the identity is true for all z = x ∈ R.

2 The Cauchy-Riemann Equations If f is any complex-valued function, then we may write f = u + iv, where u and v are real-valued functions. 16 CHAPTER 1. 1 Consider f(z) = z 2 = (x2 − y 2 ) + i(2xy); in this example u = x2 − y 2 and v = 2xy. We refer to u as the real part of f and denote it by Re f; we refer to v as the imaginary part of f and denote it by Im f. Now we formulate the notion of “holomorphic function” in terms of the real and imaginary parts of f : Let U ⊆ C be an open set and f : U → C a C 1 function.