Powers of complex numbers

13. Integer powers of complex numbers
      13.1 Modulus |z|<1
      13.2 Modulus |z|>1
      13.3 Modulus |z|=1

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13. Integer powers of complex numbers

Integer powers of complex numbers are just special cases of products. The n th power of z, written zⁿ, is equal to
zⁿ = rⁿ(cos(n)+i sin(n)),       (1.24)
where n is a positive or negative integer or zero.
If we know a complex number z, we can find zⁿ. The modulus |z|ⁿ is the n^th power of the modulus of z, the argument of zⁿ is n times the argument of z.
There are three cases: |z|<1, |z|>1, |z|ⁿ=1.
Let us consifer all of them.

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13.1 Integer powers of complex numbers. |z|<1

If |z|<1, then |z|ⁿ<|z|<1 for any integer n. The bigger n the less |z|ⁿ.

Example:
|z|= 0.9; arg(z) = 30°. Find |z|², |z|³, |z|⁴, |z|⁵, |z|⁶.

|z|²= 0.81; arg(z) = 60°;
|z|²= 0.73; arg(z) = 90°;
|z|²= 0.66; arg(z) = 120°;
|z|²= 0.59; arg(z) = 150°;
|z|²= 0.53; arg(z) = 180°.

Each higher power is 30° further along and closer to 0. Six powers are displayed as points on a spiral in the figure 1.25. This spiral is called an exponential sprial.

Figure 1.25. An exponential sprial. |z|<1.

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13.2 Integer powers of complex numbers. |z|>1

If |z|>1, then |z|ⁿ>|z|>1 for any integer n. The bigger n the bigger |z|ⁿ.

Example:
|z|= 1.2; arg(z) = 30°. Find |z|², |z|³, |z|⁴, |z|⁵, |z|⁶.

|z|²= 2.4; arg(z) = 60°;
|z|²= 2.9; arg(z) = 90°;
|z|²= 3.5; arg(z) = 120°;
|z|²= 4.2; arg(z) = 150°;
|z|²= 5; arg(z) = 180°.

Six powers are displayed as points on a spiral in the figure 1.26.

Figure 1.26. An exponential sprial. |z| >1.

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13.3 Integer powers of complex numbers. |z|=1

If |z|=1, then |z|ⁿ=1 for any integer n.

Example:
|z|= 1; arg(z) = 30°. Find |z|², |z|³, |z|⁴, |z|⁵, |z|⁶.

|z|²= 1; arg(z) = 60°;
|z|²= 1; arg(z) = 90°;
|z|²= 1; arg(z) = 120°;
|z|²= 1; arg(z) = 150°;
|z|²= 1; arg(z) = 180°.

Figure 1.27. |z| =1

Figure 1.28 shows all 3 cases

Figure 1.28.

by Tetyana Butler