Complex Analysis: Spring Semester 2020 Revision and Application
Explore the concepts of complex numbers, modulus, argument, and more in the context of complex analysis for the Spring semester of 2020. Understand the algebra of complex numbers, field properties, and work with polar form to solve problems and discuss convergence and formalism in this comprehensive study guide.
Uploaded on | 0 Views
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Complex Analysis Spring semester 2020
Revision Algebra of complex numbers ? = ? + ??, ?2= 1, ?,? ?1= ?1+ ??1,?2= ?2+ ??2 Addition ?1+ ?2= ?1+ ?2+ ?(?1+ ?2) Multiplication ?1?2= ?1?2 ?1?2+ ?(?1?2+ ?2?1)
Field of complex numbers ?1,?2 , we have Commutativity ?1+ ?2= ?2+ ?1and ?1?2= ?2?1 ?1,?2 Associativity ?1+ ?2 + ?3= ?1+ ?2+ ?3 and ?1?2?3= ?1?2?3 and Distributivity ?1?2+ ?3 = ?1?2+ ?1?3
Modulus, Argument and Complex Conjugate complex conjugate ? ? ?? modulus ? ? ? := ? ? ? ? Therefore, ? ?= ?= |?|? ?? ? :=?+ ? ?, ?? ? ? ? ?? ??(?) ??(?)argument ??? ? ??????
Polar form Euler Identity ???= cos? + ?sin ???= 1 + ?? + 1 2!?2+ ? 3!?3+ cos? = 1 + 1 2!?2+ ?sin? = ?? + ? 3!?3+ Using identity, write ? = |?|?????(?)(Polar form)
Working with polar form ?1= ?1???1,?2= ?2???2 ?1?2= ?1?2???1+??2 Multiplication Specifically, ?1?2 = ?1|?2|, arg ?1?2 = arg ?1 + arg(?2) ???? rotation of ? ????? scaling and rotation of ?
Problems Solve ?2= ? + ?? Without polar coordinates With polar coordinates Compute ? What is cos ? ? Simplify 1 + cos? + cos2? + + cos??
Convergence The sequence ?1,?2, converges to ? if lim ? ?? ? = 0 In particular, a sequence converges iff both the real parts and the imaginary parts converge. Therefore, a limit, if it exists, is unique. The sequence is Cauchy if lim ?,? |?? ??| = 0
Formalism A set is open if every point is an interior point. A set is closed if its complement is open. Open disk ??(?0) ? ? ?0 < ? Closed disk ??(?0) ? ? ?0 < ? Unit disk ?1(0) For a set , ????( ) ????,? ? ? Compactness defined as for sets in
Problems ? ? 1 ??, ?,? such that ?? 1 Show that, for ? < 1 1. ? maps the interior of ?1(0) to itself 2. ? maps the boundary of ?1(0) to itself 3. ? is bijective 4. ? 0 = ? and ? ? = 0 ? = Let ?(?) be a polynomial with real coefficients. Show that the sum of the roots is real. Show that ?? = ? |?| and use trigonometric identities to show that arg ?? = arg ? + arg(?)
