Transcription of 4. Complex integration: Cauchy integral theorem and …
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4. Complex integration: Cauchy integral theorem and …
4. Complex integration : Cauchy integral theorem and Cauchyintegral formulasDefinite integral of a Complex -valued function of a real variableConsider a Complex valued functionf(t) of a real variablet:f(t) =u(t) +iv(t),which is assumed to be a piecewise continuous function defined inthe closed intervala t b. The integral off(t) fromt=atot=b, is defined as baf(t)dt= bau(t)dt+i bav(t) of a Complex integral with real variable of baf(t)dt= baRef(t)dt= bau(t) baf(t)dt= baImf(t)dt= bav(t) ba[ 1f1(t) + 2f2(t)]dt= 1 baf1(t)dt+ 2 baf2(t)dt,where 1and 2are any Complex baf(t)dt ba|f(t)| prove (4), we consider baf(t)dt =e i baf(t)dt= bae i f(t)dt,where = Arg( baf(t)dt). Since baf(t)dt is real, we deduce that baf(t)dt = Re bae i f(t)dt= baRe [e i f(t)]dt ba|e i f(t)|dt= ba|f(t)| is real, show that|e2 i 1| 2 | |.SolutionLetf(t) =ei t, andtare real. We obtain 2 0ei tdt 2 0|ei t|dt= 2.
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable t: f(t) = u(t) + iv(t), which is assumed to be a piecewise continuous function defined in the closed interval a ≤ t ≤ b.
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