This document contains a tutorial sheet for a complex analysis course. It provides 10 practice problems covering topics such as:
1) Expressing complex numbers in different forms like rectangular and polar
2) Finding square roots of complex numbers
3) Showing relationships between the absolute value and real and imaginary parts of complex numbers
4) Showing that certain equations represent circles in the complex plane
5) Relating distances between complex numbers on the Riemann sphere
This document contains a tutorial sheet for a complex analysis course. It provides 10 practice problems covering topics such as:
1) Expressing complex numbers in different forms like rectangular and polar
2) Finding square roots of complex numbers
3) Showing relationships between the absolute value and real and imaginary parts of complex numbers
4) Showing that certain equations represent circles in the complex plane
5) Relating distances between complex numbers on the Riemann sphere
This document contains a tutorial sheet for a complex analysis course. It provides 10 practice problems covering topics such as:
1) Expressing complex numbers in different forms like rectangular and polar
2) Finding square roots of complex numbers
3) Showing relationships between the absolute value and real and imaginary parts of complex numbers
4) Showing that certain equations represent circles in the complex plane
5) Relating distances between complex numbers on the Riemann sphere
This document contains a tutorial sheet for a complex analysis course. It provides 10 practice problems covering topics such as:
1) Expressing complex numbers in different forms like rectangular and polar
2) Finding square roots of complex numbers
3) Showing relationships between the absolute value and real and imaginary parts of complex numbers
4) Showing that certain equations represent circles in the complex plane
5) Relating distances between complex numbers on the Riemann sphere
(b) Express 5e3i/4 + 2ei/6 in the form x + iy, x, y R.
2. Find all square roots of 3 + 3i.
3. Show that |z| |<e z| + |=m z|. When does equality occur? 4. Show that the equation |z|2 2<e (az) + |a|2 = r2 represents a circle centred at z = a with radius r. 5. Show that |z a| = |1 az| if |z| = 1 and a is a fixed complex number. 6. Let z0 , z1 C, and let r be a positive real number, r 6= 1. Show that the set of points z C satisfying |z z0 | = r|z z1 | is a circle. What happens when r = 1? 7. For each of the following points in C give the corresponding points on the Riemann sphere S: (i) 0; (ii) 1 + i. 8. Which subsets of the Riemann sphere S correspond to the real axis and to the imaginary axis in C? 9. Let Z and Z 0 be the points on the Riemann sphere S corresponding to the points z and z 0 in C, respectively. Let W be the point on S corresponding to the point z + z 0 in C. Find the coordinates of W in terms of the coordinates of Z and Z 0 . 10. Show that z and z 0 in C correspond to diametrically opposite points on the Riemann sphere S if and only if zz 0 = 1.