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2.4. VISUALIZING COMPLEX FUNCTIONS 33 2.3.2 EXERCISES 2.3. Sketch the following lines and circles and their images under the given mapping $f$. (a) $z(t) = 1 + it$, $t \in \mathbb{R}$, $f(z) = iz$. (b) $iz - \overline{iz} = 2$, $f(z) = (-2\sqrt{3} + 2i)z - 2 - i$. (c) $z(t) = -2 - 2i + 4e^{it}$, $0 \le t < 2\pi$, $f(z) = -z - 2$. 2.4. What is the center and radius of the circle obtained by mapping the line $(3 + 4i)z +$ $(3 - 4i)\overline{z} = 4$ by the inversion $f(z) = 1/z$? [Hint: Put the line formula in standard form first by scaling so that $|A| = 1$.] 2.5. What is the distance to the origin and the normal to the line obtained by mapping the circle $z\overline{z} - (2 + 3i)z - (2 - 3i)\overline{z} = 0$ by the inversion $f(z) = 1/z$? 2.6. Show that the effect of inversion by $f(z) = 1/z$ on stereographic projection is a re- flection across the $y = 0$ plane and across the $z = 0$ plane for points on the Riemann sphere. That is, if $z$ is stereographically projected to $A = (X, Y, Z)$ on the Riemann sphere, then $w = 1/z$ is stereographically projected to $B = (X, Y, -Z)$. [Hint: Use the result of Exercise 1.23.]

          2.4. VISUALIZING COMPLEX FUNCTIONS 33
2.3.2 EXERCISES
2.3. Sketch the following lines and circles and their images under the given mapping $f$.
(a) $z(t) = 1 + it$, $t \in \mathbb{R}$, $f(z) = iz$.
(b) $iz - \overline{iz} = 2$, $f(z) = (-2\sqrt{3} + 2i)z - 2 - i$.
(c) $z(t) = -2 - 2i + 4e^{it}$, $0 \le t < 2\pi$, $f(z) = -z - 2$.
2.4. What is the center and radius of the circle obtained by mapping the line $(3 + 4i)z +$
$(3 - 4i)\overline{z} = 4$ by the inversion $f(z) = 1/z$? [Hint: Put the line formula in standard
form first by scaling so that $|A| = 1$.]
2.5. What is the distance to the origin and the normal to the line obtained by mapping the
circle $z\overline{z} - (2 + 3i)z - (2 - 3i)\overline{z} = 0$ by the inversion $f(z) = 1/z$?
2.6. Show that the effect of inversion by $f(z) = 1/z$ on stereographic projection is a re-
flection across the $y = 0$ plane and across the $z = 0$ plane for points on the Riemann
sphere. That is, if $z$ is stereographically projected to $A = (X, Y, Z)$ on the Riemann
sphere, then $w = 1/z$ is stereographically projected to $B = (X, Y, -Z)$. [Hint: Use
the result of Exercise 1.23.]

2.4. VISUALIZING COMPLEX FUNCTIONS 33
2.3.2 EXERCISES
2.3. Sketch the following lines and circles and their images under the given mapping f.
(a) z(t) = 1 + it, t ∈ℝ, f(z) = iz.
(b) iz - iz = 2, f(z) = (-2√(3) + 2i)z - 2 - i.
(c) z(t) = -2 - 2i + 4e^it, 0 ≤ t < 2π, f(z) = -z - 2.
2.4. What is the center and radius of the circle obtained by mapping the line (3 + 4i)z +
(3 - 4i)z = 4 by the inversion f(z) = 1/z? [Hint: Put the line formula in standard
form first by scaling so that |A| = 1.]
2.5. What is the distance to the origin and the normal to the line obtained by mapping the
circle zz - (2 + 3i)z - (2 - 3i)z = 0 by the inversion f(z) = 1/z?
2.6. Show that the effect of inversion by f(z) = 1/z on stereographic projection is a re-
flection across the y = 0 plane and across the z = 0 plane for points on the Riemann
sphere. That is, if z is stereographically projected to A = (X, Y, Z) on the Riemann
sphere, then w = 1/z is stereographically projected to B = (X, Y, -Z). [Hint: Use
the result of Exercise 1.23.]

Added by Bradley C.

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