3 Complex Integration - ADVANCED ENGINEERING MATHEMATICS...

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ADVANCED ENGINEERING MATHEMATICS
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Complex Integration As in calculus we distinguish between definite integrals and indefinite integrals. An indefinite integral is a function whose derivative equal a given analytic function in a region. By inverting known differentiation formulas we may find many types of indefinite integrals. Complex definite integrals are called (complex) line integrals. •They are written as 𝐶 ? ? ??
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Complex Integration The integrand ? ? is integrated over a given curve C or a portion of it. This curve C in the complex plane is called the path of integration. We may represent C by a parametric equation ? 𝑡 = ? 𝑡 + 𝑖? 𝑡 Example: The circle ? = 4 can be represented by ? 𝑡 = 4 cos 𝑡 + 4𝑖 sin 𝑡 𝜋 𝑡 𝜋 The line ? = 3? can be represented by ? 𝑡 = 𝑡 + 3𝑖𝑡 0 𝑡 2
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Complex Integration General Assumption: All paths of integration for complex line integrals are assumed to be piecewise smooth, that is, they consist of finitely many smooth curves joined end to end.
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