Answer to Question #102646 in Complex Analysis for BIVEK SAH

Question #102646
Evaluate the integral

Integral [(5z^2-3z+2)]/[(z-1)^3] dz
Close curve C



where C is any simple closed curve enclosing z = 1.
1
Expert's answer
2020-02-19T10:44:11-0500

By the condition of the problem it is necessary to find the integral "\\oint_c \\frac{5z^2-3z+2}{(z-1)^3}dz" 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) "\\oint_{\\gamma} \\frac{5(w+1)^2-3(w+1)+2}{w^3}dw"

where "\\gamma" is any contour enclosing point "w=0". Simplify numerator we have

(2) "5(w+1)^2-3(w+1)+2=5(w^2+2w+1)-3w-3+2=5w^2+7w+4"

and

(3)"\\oint_{\\gamma} \\frac{5w^2+7w+4}{w^3}dw"

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\\cdot e^{i\\phi}" where "r,\\phi \\in R_1; r=const; \\phi \\in [0,2\\pi]". We rewrite integral (3) as

(4) "\\oint_{\\gamma} \\frac{5w^2+7w+4}{w^3}dw=\\oint_{\\gamma} \\frac{5}{w}dw+\\oint_{\\gamma} \\frac{7}{w^2}dw+\\oint_{\\gamma} \\frac{4}{w^3}dw=\\\\=5i \\int^{2\\pi}_{0} d\\phi+7i \\int^{2\\pi}_{0} \\frac{1}{re^{i\\phi}}d\\phi+4i \\int^{2\\pi}_{0} \\frac{1}{(re^{i\\phi})^2}d\\phi"

We given that "dw=r\\cdot i\\cdot e^{i\\phi}d\\phi" .

The second and third integrals in (4) equals 0. For example second integral is

"\\int^{2\\pi}_{0} \\frac{1}{re^{i\\phi}}d\\phi =\\frac{1}{r} \\int^{2\\pi}_{0} e^{-i\\phi}d\\phi=\\frac{1}{r}(\\frac{e^{-i\\phi}}{-i})^{2\\pi}_0=\\frac{i}{r}(e^{-i2\\pi}-e^{0})=\\frac{i}{r}(1-1)=0"

The first one is

"5i \\int^{2\\pi}_{0} d\\phi=10\\pi i"

Answer: The Integral "\\oint_c \\frac{5z^2-3z+2}{(z-1)^3}dz=10\\pi i"

where C is any simple closed curve enclosing z = 1.



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