Let n eN (that is, n is a positive whole number). The n-th root of a complex number z is z1/n. When dealing with roots we have to be careful since the n-th root function is multi-valued. In particular, there are n different possible values of w such that w" = z, all of which could properly be considered n-th roots of z! Let's explore this! ein/2. Show that eir/6, e5in/6, and e3it/2 (g) Consider the cube roots of i of i. Plot these roots on the complex plane and demonstrate graphically how ešin/6 cubes to i. [Note: Remember that multiplying by a phase e?º essentially "rotates" a point in the complex plane counterclockwise by an angle 0.] are all cube roots Find a formula for the n distinct values of the n-th root of z = rei®. As a check, test that your (h) formula implies that if w is a square root of z then so is -w.
Let n eN (that is, n is a positive whole number). The n-th root of a complex number z is z1/n. When dealing with roots we have to be careful since the n-th root function is multi-valued. In particular, there are n different possible values of w such that w" = z, all of which could properly be considered n-th roots of z! Let's explore this! ein/2. Show that eir/6, e5in/6, and e3it/2 (g) Consider the cube roots of i of i. Plot these roots on the complex plane and demonstrate graphically how ešin/6 cubes to i. [Note: Remember that multiplying by a phase e?º essentially "rotates" a point in the complex plane counterclockwise by an angle 0.] are all cube roots Find a formula for the n distinct values of the n-th root of z = rei®. As a check, test that your (h) formula implies that if w is a square root of z then so is -w.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let n eN (that is, n is a positive whole number). The n-th root of a complex number z is z1/n.
When dealing with roots we have to be careful since the n-th root function is multi-valued. In
particular, there are n different possible values of w such that w" = z, all of which could properly
be considered n-th roots of z! Let's explore this!
ein/2. Show that eir/6, e5in/6, and e3it/2
(g) Consider the cube roots of i
of i. Plot these roots on the complex plane and demonstrate graphically how ešin/6 cubes to i.
[Note: Remember that multiplying by a phase e?º essentially "rotates" a point in the complex plane
counterclockwise by an angle 0.]
are all cube roots
Find a formula for the n distinct values of the n-th root of z = rei®. As a check, test that your
(h)
formula implies that if w is a square root of z then so is -w.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e89d9d-bfd3-4a29-98a6-3c58c69fa30f%2Fd7a51dc6-1764-4505-b53b-bcb2a4c780be%2Fmt1ft6p.png&w=3840&q=75)
Transcribed Image Text:Let n eN (that is, n is a positive whole number). The n-th root of a complex number z is z1/n.
When dealing with roots we have to be careful since the n-th root function is multi-valued. In
particular, there are n different possible values of w such that w" = z, all of which could properly
be considered n-th roots of z! Let's explore this!
ein/2. Show that eir/6, e5in/6, and e3it/2
(g) Consider the cube roots of i
of i. Plot these roots on the complex plane and demonstrate graphically how ešin/6 cubes to i.
[Note: Remember that multiplying by a phase e?º essentially "rotates" a point in the complex plane
counterclockwise by an angle 0.]
are all cube roots
Find a formula for the n distinct values of the n-th root of z = rei®. As a check, test that your
(h)
formula implies that if w is a square root of z then so is -w.
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