Transcription of Lecture Notes for Complex Variables
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Lecture Notes for Complex Variables
Lecture Notes for Complex VariablesJames S. CookLiberty UniversityDepartment of Mathematics and PhysicsSpring 20102introduction and motivations for these notesA Complex variable is simply a variable whose possible values are allowed to reside in the complexnumbers. We re using the classic text by Churchill and Brown: Complex Variables and Applications by Churchill and Brown, 6-th text has been a staple of several generations of mathematicians at this time. I ll try to followthe text somewhat closely. I plan to ask you to prove certain pivotal Lemmas as we develop thematerial together this semester. In previous courses you may have heard me advocate a certainpoint of view about Complex numbers but I would ask you forget all that for a time. Our goal hereis to start from scratch and build Complex numbers from the ground up. The purpose of thesenotes is to complement Churchill s text.
Xcomplex logarithms and subtletites of multiply valued functions. Xtopological properties of C. Xcontinuous functions of a complex variable. Xcomplex di erentiation and the Cauchy Riemann equations. Xthe conjugate variable notation, homomorphic and antiholomorphic. XMaximum modulus theorem. XCauchy-Goursat theorem. Xcontour integration ...
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