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 | |.
Cauchy integral theorem Let f(z) = u(x,y)+iv(x,y) be analytic on and inside a simple closed contour C and let f′(z) be also continuous on and inside C, then I C f(z) dz = 0. Proof The proof of the Cauchy integral theorem requires the Green theo-
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