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Question Number 213604 by issac last updated on 10/Nov/24
show that  ∫_C  e^z^3   dz=0  where C is any simple closed contour.        Evaluate the integral  ∫_( C_1 ) f(z)dz , ∫_( C_2 ) f(z)dz  where f(z)=(y−x)−3x^2 i  C_3 =OA ; z(y)=x+iy=iy , (0≤y≤1)  C_1 =AB ; z(x)=x+iy=x+i , (0≤x≤1)  C_2 =OB ; z(x)=x+iy=x+ix , (0≤x≤1)     Let′s C be the quadrant  z=2e^(iθ)  ,0≤θ≤(π/2)  show that   ∣∫_( C)  ((z^  +4)/(z^3 −1)) dz∣≤((6π)/7)     Let C be any simple closed contour  described in the positive sense in the z plane  and write  g(z)=∫_( C)   ((s^3 +2s)/((z−2s)^3 )) ds  show that g(z)=6πiz when z is inside C   show that g(z)=0 when z is outside C
showthat\boldsymbolCez3dz=0where\boldsymbolCisanysimpleclosedcontour.EvaluatetheintegralC1f(z)dz,C2f(z)dzwheref(z)=(yx)3x2\boldsymboliC3=OA;z(y)=x+\boldsymboliy=\boldsymboliy,(0y1)C1=AB;z(x)=x+\boldsymboliy=x+\boldsymboli,(0x1)C2=OB;z(x)=x+\boldsymboliy=x+\boldsymbolix,(0x1)LetsCbethequadrantz=2e\boldsymboliθ,0θπ2showthatCz+4z31dz∣≤6π7LetCbeanysimpleclosedcontourdescribedinthepositivesenseinthezplaneandwriteg(z)=Cs3+2s(z2s)3dsshowthatg(z)=6π\boldsymbolizwhenzisinsideCshowthatg(z)=0whenzisoutsideC

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