By Steven G. Krantz
This is a ebook approximately advanced variables that provides the reader a short and available creation to the main themes. whereas the assurance isn't really entire, it definitely supplies the reader a great grounding during this basic quarter. there are various figures and examples to demonstrate the important principles, and the exposition is vigorous and welcoming. An undergraduate eager to have a primary examine this topic or a graduate scholar getting ready for the qualifying tests, will locate this ebook to be an invaluable source.
In addition to big rules from the Cauchy thought, the booklet additionally comprise sthe Riemann mapping theorem, harmonic features, the argument precept, common conformal mapping and dozens of different important topics.
Readers will locate this booklet to be an invaluable better half to extra exhaustive texts within the box. it's a worthwhile source for mathematicians and non-mathematicians alike.
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Extra info for A Guide to Complex Variables
Let f be a holomorphic function on U. 1) γ does not change if the curve γ is smoothly deformed within the region U. Note that, in order for this statement to be valid, the curve γ must remain inside the region of holomorphicity U of f while it is being deformed, and it must remain a closed curve while it is being deformed. 5 shows curves γ1 , γ2 that can be deformed to one another, and a curve γ3 that can be deformed to neither of the first two (because of the hole inside γ3 ). 4 A Coda on the Limitations of The Cauchy Integral Formula If f is any continuous function on the boundary of the unit disc D = D(0, 1), then the Cauchy integral F (z) = 1 2πi ∂D f(ζ) dζ ζ −z defines a holomorphic function F (z) on D (use Morera’s theorem, for example, to confirm this assertion).
In order to work effectively with γ we need to impose on it some differentiability properties. 3) ϕ has a continuous extension to [a, b]. 1. 2: A simple, closed curve. In other words, we require that lim ϕ (t) and lim ϕ (t) t→b− t→a+ both exist. 4) a so that the Fundamental Theorem of Calculus holds for ϕ ∈ C 1 ([a, b]). A curve γ : [a, b] → C, with γ(t) = γ1 (t)+iγ2(t) is said to be continuous on [a, b] if both γ1 and γ2 are. We write γ ∈ C 0([a, b]). 5) if γ1 , γ2 are continuously differentiable on [a, b].
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. Write f(z) = f(x + iy) ≡ f(x, y) = u(x, y) + iv(x, y), with z = x + iy and u and v real-valued functions. 4, where a formal definition of “holomorphic” is provided). 2) are called the Cauchy-Riemann equations. A practical method for checking whether a given function is holomorphic is to check whether it satisfies the Cauchy-Riemann equations.
A Guide to Complex Variables by Steven G. Krantz