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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.
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.