Let f(z) = u + iv = z7+6z3+7
Let z = Reiθ, and 0≤θ≤π2, R→∞
f(z) is analytic at all points except z = ∞. Therefore, it is analytic within and upon the complementary of first quadrant.
when z = x,
f(z)=u+iv=x7+6x3+7
argf=tan−1(vu)=tan−1(0x7+6x3+7) = 0, ∀ x ≥ 0
Therefore, argf=0
when z = Reiθ, 0≤θ≤π2, R→∞
f(z) = R7e7iθ(1+6R4e4iθ+7R7e7iθ)
when R→∞, f→R7e7iθ and arg f = 7θ
argf=7(π2−0)=7π2
when z = iy,
f(z) = u + iv =7+6x3+7
argf=tan−1(vu)=tan−1(y7−6y37)=π2 from ∞→0
argf=7π2+π2=4π
Thus, the angle change is 4π, and the number of zero in the first quadrant is 2.