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# A Guide to Complex Variables by Steven G. Krantz

By Steven G. Krantz

This is a e-book approximately complicated variables that provides the reader a short and obtainable creation to the foremost subject matters. whereas the assurance isn't entire, it definitely supplies the reader a fantastic grounding during this primary quarter. there are numerous figures and examples to demonstrate the relevant principles, and the exposition is full of life and alluring. An undergraduate desirous to have a primary examine this topic or a graduate pupil getting ready for the qualifying checks, will locate this booklet to be an invaluable source.

In addition to big principles from the Cauchy thought, the e-book additionally contain sthe Riemann mapping theorem, harmonic capabilities, the argument precept, basic conformal mapping and dozens of different critical topics.

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

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In other words, for such a function u there exists a harmonic function v defined on U such that f ≡ u + iv is holomorphic on U. Any two such functions v must differ by a constant. We call the function v a harmonic conjugate for u. To give an indication of why these statements are true we note that, given u harmonic, we seek v such that ∂v ∂u =− ≡ α(x, y) ∂x ∂y and ∂v ∂u = ≡ β(x, y) ∂y ∂x (these are the Cauchy-Riemann equations). We know from calculus that a pair of equations like this is solvable on a region with no holes precisely when ∂α ∂β = .

A converse is true provided the functions involved are defined on a domain with no holes: 22 CHAPTER 1. THE COMPLEX PLANE Theorem: If R is an open rectangle (or open disc) and if u is a real-valued harmonic function on R, then there is a holomorphic function F on R such that Re F = u. In other words, for such a function u there exists a harmonic function v defined on R such that f ≡ u + iv is holomorphic on R. Any two such functions v must differ by a real constant. 3), and if u is harmonic on U, then there is a holomorphic function F on U with Re F = u.

Plane. 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).

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