Final answer:
The argument principle combined with Rouche's theorem can be used to show that f(z)=z⁵ +1 has only one zero and f(z)=z⁷ +1 has two zeroes in the first quadrant, by analyzing the change in the argument of the functions along appropriately chosen contours.
Step-by-step explanation:
The student is asking how to show that the function f(z)=z⁵ +1 has only one zero in the first quadrant and that the function f(z)=z⁷ +1 has two zeroes in the first quadrant using the argument principle.
To demonstrate this for f(z)=z⁵ +1, we must consider a contour that encloses the first quadrant and then compute the number of times the function's argument changes as we traverse this contour. Since all the coefficients of the polynomial are positive and there is a constant term, Rouche's theorem can be used in conjunction with the argument principle to show there is only one root within the contour that passes through the first quadrant.
For the function f(z)=z⁷ +1, a similar approach is taken, but in this case, we will find that the change in argument indicates that there are two zeros in the first quadrant. This can be shown by choosing an appropriate contour and applying the argument principle and Rouche's theorem.
Note that to apply these theorems correctly, the contours must not pass through any of the zeros of the functions, and the chosen contours should be large enough that the dominant term will determine the change in argument.