By the condition of the problem it is necessary to find the integral ∮c(z−1)35z2−3z+2dz along the contour c enclosing the point z=1. For simplicity we will make a replacement z=w+1 then dz=dw and integral become
(1) ∮γw35(w+1)2−3(w+1)+2dw
where γ is any contour enclosing point w=0. Simplify numerator we have
(2) 5(w+1)2−3(w+1)+2=5(w2+2w+1)−3w−3+2=5w2+7w+4
and
(3)∮γw35w2+7w+4dw
Note that the integrand function is unambiguous, differentiable, and continuous in the entire area surrounded by the contour except for the point . This means that it is an analytic function in the specified area with the exception of point w=0 . In the theory of analytic functions, it is proved that any change in the contour in the region of analyticity of a function does not change the value of the integral of this function along the contour. Then we can integrate about some circle, with w=r⋅eiϕ where r,ϕ∈R1;r=const;ϕ∈[0,2π]. We rewrite integral (3) as
(4) ∮γw35w2+7w+4dw=∮γw5dw+∮γw27dw+∮γw34dw==5i∫02πdϕ+7i∫02πreiϕ1dϕ+4i∫02π(reiϕ)21dϕ
We given that dw=r⋅i⋅eiϕdϕ .
The second and third integrals in (4) equals 0. For example second integral is
∫02πreiϕ1dϕ=r1∫02πe−iϕdϕ=r1(−ie−iϕ)02π=ri(e−i2π−e0)=ri(1−1)=0
The first one is
5i∫02πdϕ=10πi
Answer: The Integral ∮c(z−1)35z2−3z+2dz=10πi
where C is any simple closed curve enclosing z = 1.
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