Transcription of Complex variable solvedproblems - Univerzita Karlova
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Complex variable solvedproblems - Univerzita Karlova
Complex variablesolved problemsPavel Pyrih11:03 May 29, 2012( public domain )Contents1 Residue theorem problems22 Zero Sum theorem for residues problems763 Power series problems157 Acknowledgement. The following problems were solved using my own procedurein a program Maple V, release 5. All possible errors are my Residue theorem problemsWe will solve several problems using the following theorem:Theorem.(Residue theorem) SupposeUis a simply connected open subsetof the Complex plane, andw1, .. ,wnare finitely many points ofUand f is afunction which is defined and holomorphic onU\{w1,..,wn}. If is a simplyclosed curve inUcontaning the pointswkin the interior, then f(z)dz= 2 ik k=1res (f,wk).The following rules can be used for residue counting:Theorem.
We will use special formulas for special types of problems: Theorem. ( TYPE I. Integral from a rational function in sin and cos.) If Q(a;b) is a rational function of two complex variables such that for real a;b,
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