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### Messages - sishan

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##### Quiz-7 / Re: Q7 TUT 0202
« on: December 01, 2018, 03:11:55 AM »
Let f(z) = u + iv = $z^7 + 6z^3 +7$

Let z = $Re^{i\theta}$, and  $0\leq \theta \leq \frac{\pi}{2}$,  $R\to \infty$

f(z) is analytic at all points except z = $\infty$. Therefore, it is analytic within and upon the complementary of first quadrant.

when z = x,

$f(z) = u + iv = x^7 + 6x^3 + 7$

$arg f = tan^{-1}(\frac{v}{u}) = tan^{-1}(\frac{0}{x^7 + 6x^3 + 7})$ = 0, $\forall$ x $\geq$ 0

Therefore, $arg f = 0$

when z = $Re^{i\theta}$, $0\leq \theta \leq \frac{\pi}{2}$,  $R\to \infty$

f(z) = $R^7e^{7i\theta}(1+\frac{6}{R^4e^{4i\theta}} + \frac{7}{R^7e^{7i\theta}})$

when $R\to \infty$, $f \to R^7e^{7i\theta}$  and arg f = $7\theta$

$argf = 7(\frac\pi2-0) = \frac{7\pi}{2}$

when z = iy,

f(z) = u + iv =$^7 + 6x^3 + 7$

$argf = tan^{-1}(\frac{v}{u})= tan^{-1}(\frac{y^7-6y^3}{7}) = \frac{\pi}{2}$  from $\infty \to 0$

$argf = \frac{7\pi}{2}+\frac{\pi}{2} = 4\pi$

Thus, the angle change is $4\pi$, and the number of zero in the first quadrant is 2.

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