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Residue integration 01
1.
Residue integration Hanpen
Robot 2015/02/12 ★Formula 01 ∫ 𝑓(𝑒 𝑗𝜃 2𝜋 0 )𝑑𝜃 = ∫ 𝑓(𝑧) 𝑑𝑧 𝑗𝑧𝐶 , (𝐶: 𝑧(𝑡) = 𝑒 𝑗𝜃 , 0 ≤ 𝜃 < 2𝜋) ∴ ∫ 𝑓(𝑒 𝑗𝜃 2𝜋 0 )𝑑𝜃 = ∑ Res [ 𝑓(𝑧) 𝑗𝑧 , 𝛼𝑖] Where 𝑓(𝑧) is rational function. ★Proof: Obvious by definition of a line integral. ★Problem 01: Calculate the definite integral of the following: ∫ cos2𝑛(𝜃) 𝜋 0 𝑑𝜃, (𝑛 = 1,2,3, … ) ★Answer: Using Formula 01, ∫ cos2𝑛(𝜃) 𝜋 0 𝑑𝜃 = 1 2 ∫ cos2𝑛(𝜃) 2𝜋 0 𝑑𝜃 = 1 2 ∫ ( 𝑧 + 𝑧−1 2 ) 2𝑛 𝑑𝑧 𝑗𝑧𝐶 = 1 𝑗22𝑛+1 ∫ (𝑧 + 1 𝑧 ) 2𝑛 𝑑𝑧 𝑧𝐶 = 1 𝑗22𝑛+1 ∫ ( 𝑧2 + 1 𝑧 ) 2𝑛 𝑑𝑧 𝑧𝐶 = 1 𝑗22𝑛+1 ∫ (𝑧2 + 1)2𝑛 𝑧2𝑛+1 𝑑𝑧 𝐶 = 1 𝑗22𝑛+1 ∫ 1 𝑧2𝑛+1 ∑ ( 2𝑛 𝑘 ) 𝑧2(2𝑛−𝑘) 2𝑛 𝑘=0 𝑑𝑧 𝐶 = 1 𝑗22𝑛+1 ∫ 1 𝑧2𝑛+1 ∑ ( 2𝑛 𝑘 ) 𝑧4𝑛−2𝑘 2𝑛 𝑘=0 𝑑𝑧 𝐶 = 1 𝑗22𝑛+1 ∫ ∑ ( 2𝑛 𝑘 ) 𝑧4𝑛−2𝑘−2𝑛−1 2𝑛 𝑘=0 𝑑𝑧 𝐶 = 1 𝑗22𝑛+1 ∫ ∑ ( 2𝑛 𝑘 ) 𝑧2𝑛−2𝑘−1 2𝑛 𝑘=0 𝑑𝑧 𝐶 = 1 𝑗22𝑛+1 ∫ ( 2𝑛 𝑛 ) 𝑧−1 𝑑𝑧 𝐶 = 1 𝑗22𝑛+1 ( 2𝑛 𝑛 ) ∫ 𝑑𝑧 𝑧𝐶 = 1 𝑗22𝑛+1 ( 2𝑛 𝑛 ) 2𝜋𝑗 = 𝜋 22𝑛 ( 2𝑛 𝑛 ) ∴ ∫ cos2𝑛(𝜃) 𝜋 0 𝑑𝜃 = 𝜋 22𝑛 ( 2𝑛 𝑛 )
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