# lecture3.pdf - NPTEL web course on Complex Analysis A...

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NPTEL web courseonComplex AnalysisA. SwaminathanI.I.T. Roorkee, IndiaandV.K. KatiyarI.I.T. Roorkee, IndiaA.Swaminathan and V.K.Katiyar (NPTEL)Complex Analysis1 / 20
Complex AnalysisModule: 1:IntroductionLecture: 3:Inequalities and complex exponentsA.Swaminathan and V.K.Katiyar (NPTEL)Complex Analysis2 / 20
Complex Number systemTriangle inequalityA.Swaminathan and V.K.Katiyar (NPTEL)Complex Analysis3 / 20
Geometry of complex numbersProperty of a triangleIn any triangle, sum of lengths of any two sides is more than that of thethird side. We see an analogous result in the complex plane.A.Swaminathan and V.K.Katiyar (NPTEL)Complex Analysis4 / 20
Geometry of complex numbersTriangle inequalityLetz1andz2be any two complex numbers. Then|z1+z2|2= (z1+z2)(z1+z2)= (z1+z2)(z1+z2)=z1z1+z1z2+z2z1+z2z2=|z1|2+|z2|2+z1z2+z1z2(z=z)=|z1|2+|z2|2+2Rez1z2(z+z=2Rez)≤ |z1|2+|z2|2+2|z1z2|(Rez≤ |z|)=|z1|2+|z2|2+2|z1||z2|(|z1z2|=|z1||z2|)=|z1|2+|z2|2+2|z1||z2|(|z|=|z|=r)= (|z1|+|z2|)2.A.Swaminathan and V.K.Katiyar (NPTEL)Complex Analysis5 / 20
Geometry of complex numbersTriangle inequalityThis implies|z1+z2|2(|z1|+|z2|)
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