Use polar forms, arguments and complex multiplication geometry

Powers before roots

Use the polar multiplication identity to calculate integer powers. Prove the rule by induction, then reverse the modulus and angle equations to find every root

Integer powers repeat the multiplication rule

Apply the integer-power rule to the radius and angle. A negative or zero exponent requires a nonzero base

1. 1Power rule The displayed polar form uses a positive radius \(\displaystyle \begin{gathered}z=r\operatorname{cis}\theta,\quad r>0,\ n\in\mathbb Z\\z^n=r^n\operatorname{cis}(n\theta)\end{gathered}\)
2. 2Principal argument Multiply the argument first, then add whole turns to enter the principal interval \(\displaystyle -\pi<\operatorname{Arg}(z^n)\le\pi\)
3. 3Zero base Only positive integer powers of zero are defined here \(\displaystyle 0^n=0\quad(n\in\mathbb Z_{>0})\)

Worked reciprocal power

Find the polar data of the base, apply the negative exponent, then convert back to Cartesian form

1. 1Base The point lies in the first quadrant \(\displaystyle \begin{gathered}z=\sqrt3+i\\|z|=\sqrt{3+1}=2,\quad\operatorname{Arg}(z)=\frac\pi6\end{gathered}\)
2. 2Reciprocal square Raise the radius and multiply the angle \(\displaystyle \begin{gathered}z^{-2}=2^{-2}\operatorname{cis}\left(-\frac{2\pi}{6}\right)\\=\frac14\operatorname{cis}\left(-\frac\pi3\right)\end{gathered}\)
3. 3Cartesian components Use the exact cosine and sine \(\displaystyle \begin{aligned}z^{-2}&=\frac14\left(\frac12-\frac{\sqrt3}{2}i\right)\\&=\frac18-\frac{\sqrt3}{8}i\end{aligned}\)

Base

\(\displaystyle z=\sqrt3+i\)

Power

\(\displaystyle z^{-2}\)

Enter the modulus of the reciprocal power

Base

\(\displaystyle z=\sqrt3+i\)

Power

\(\displaystyle z^{-2}\)

Enter the coefficient of pi in the principal argument

Prove the multiplication and power rules, ungraded

Use sine and cosine addition identities to prove the multiplication rule. Then prove the integer-power rule for positive, zero and negative integers, stating the zero-base restrictions

Model: addition identities and induction

Expand using the sine and cosine addition identities. Induct on positive integers by multiplying by one more cis factor. For a nonzero base, the zero power is one and reciprocals give negative powers. Apply the real exponent laws to the radius. Positive powers of zero are zero; the other zero-base cases are excluded

1. 1Addition identities Compare real and imaginary parts \(\displaystyle \operatorname{cis}\alpha\operatorname{cis}\beta=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta)=\operatorname{cis}(\alpha+\beta)\)
2. 2Reciprocals Use the inverse angle \(\displaystyle \operatorname{cis}\theta\operatorname{cis}(-\theta)=1\)

Make the induction hypothesis precise

Fix a real angle and define the statement for each positive integer. The base case follows directly. Assume the statement for one arbitrary positive integer, then multiply by one more cis factor and use the multiplication identity. This proves the positive-integer statement

1. 1Induction statement For every positive integer \(\displaystyle P(n):\ (\operatorname{cis}\theta)^n=\operatorname{cis}(n\theta),\quad n\in\mathbb Z_{>0}\)
2. 2Induction step Assume the statement for an arbitrary positive integer \(\displaystyle (\operatorname{cis}\theta)^{k+1}=(\operatorname{cis}\theta)^k\operatorname{cis}\theta=\operatorname{cis}(k\theta)\operatorname{cis}\theta=\operatorname{cis}((k+1)\theta),\quad k\ge1\)

Extend the proof with valid inverses

For a nonzero base with positive radius, the zero power is one. For a negative exponent, take the reciprocal of the corresponding positive power and use the inverse angle. Positive powers of zero are zero, but zero has no polar direction and its nonpositive powers are excluded. This supplies the complex-power induction application; later vector/calculus applications remain separate

1. 1Zero power The base is nonzero \(\displaystyle z=r\operatorname{cis}\theta,\quad r>0,\qquad z^0=1=r^0\operatorname{cis}0\)
2. 2Negative power Take a positive integer exponent and its reciprocal \(\displaystyle n=-m,\quad m\in\mathbb Z_{>0},\qquad z^{-m}=\frac{1}{r^m\operatorname{cis}(m\theta)}=r^{-m}\operatorname{cis}(-m\theta)\)
3. 3Inverse angle The cis factors multiply to one \(\displaystyle \operatorname{cis}(m\theta)\operatorname{cis}(-m\theta)=1\)

In the induction step, which statement may be assumed?