By Steven G. Krantz
This is a ebook approximately advanced variables that offers the reader a brief and obtainable creation to the main themes. whereas the insurance isn't really accomplished, it definitely supplies the reader a great grounding during this primary sector. there are numerous figures and examples to demonstrate the primary principles, and the exposition is energetic and alluring. 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 special rules from the Cauchy conception, the booklet additionally comprise sthe Riemann mapping theorem, harmonic services, the argument precept, normal conformal mapping and dozens of alternative valuable topics.
Readers will locate this ebook to be an invaluable spouse to extra exhaustive texts within the box. it's a useful source for mathematicians and non-mathematicians alike.
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Additional info for A Guide to Complex Variables
More generally, if U is a region with no holes (a simply connected region),1 and if u is harmonic on U , then there is a holomorphic function F on U with Re F D u. Put differently, for such a function u there exists a harmonic function v defined on U such that f Á u C 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. 1 We say that a domain U Â C is simply connected if any closed curve in U can be continuously deformed to a point.
P; r /. P; r /, then let f denote the function to which the power series converges. z/ ! P; r 0 /. P; r /. z/ ! 3:1:6:4/ This shows that a differentiated power series has a disc of convergence at least as large as the disc of convergence (with the same center) of the original series, and that the differentiated power series converges on that disc to the derivative of the sum of the original series. P; r /. j / on f denotes the j th derivative. The provenance of this formula will be explained below.
P; r /. z/ ! 3:1:6:4/ This shows that a differentiated power series has a disc of convergence at least as large as the disc of convergence (with the same center) of the original series, and that the differentiated power series converges on that disc to the derivative of the sum of the original series. P; r /. j / on f denotes the j th derivative. The provenance of this formula will be explained below. Thus we have an explicit way of calculating the power series expansion of any holomorphic function f about a point P of its domain, and we have an a priori knowledge of the disc on which the power series representation will converge.
A Guide to Complex Variables by Steven G. Krantz