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Let-f-is-a-2-nd-degree-polynomial-1-2pii-z-2-z-f-z-f-z-dz-0-and-1-2pii-z-2-z-2-f-z-f-z-dz-2-If-f-0-2017-find-explicit-form-of-f-z-




Question Number 52499 by Joel578 last updated on 09/Jan/19
Let f is a 2^(nd)  degree polynomial.  (1/(2πi)) ∫_(∣z∣=2) ((z f ′(z))/(f(z))) dz = 0,  and (1/(2πi)) ∫_(∣z∣=2) ((z^2  f ′(z))/(f(z))) dz = −2  If f(0) = 2017, find explicit form of f(z)
$$\mathrm{Let}\:{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{degree}\:\mathrm{polynomial}. \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\int_{\mid{z}\mid=\mathrm{2}} \frac{{z}\:{f}\:'\left({z}\right)}{{f}\left({z}\right)}\:{dz}\:=\:\mathrm{0},\:\:\mathrm{and}\:\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\int_{\mid{z}\mid=\mathrm{2}} \frac{{z}^{\mathrm{2}} \:{f}\:'\left({z}\right)}{{f}\left({z}\right)}\:{dz}\:=\:−\mathrm{2} \\ $$$$\mathrm{If}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{2017},\:\mathrm{find}\:\mathrm{explicit}\:\mathrm{form}\:\mathrm{of}\:{f}\left({z}\right) \\ $$

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