Exponential Form

Exponential Form

It is known (and can be proved using special infinite series — a deeper result not derived here) that

This is Euler's identity. Combining it with the polar form (Section 1.5.4):

where $r=|z|$ and $\theta=\arg z$, this is called the exponential form of the complex number.

Continuing the previous worked example, the same four numbers converted to exponential form:

1. $z=4+4\sqrt3i=8e^{i\pi/3}$.
2. $z=-2=2e^{i\pi}$.
3. $z=3i=3e^{i\pi/2}$.
4. $z=-\sqrt3+i=2e^{5\pi i/6}$. Worked Example: express $z=\sqrt2\,e^{3\pi i/4}$ in $a+ib$ form. Here $r=\sqrt2,\ \theta=\dfrac{3\pi}{4}$. The polar form is $z=\sqrt2\left(\cos\dfrac{3\pi}{4}+i\sin\dfrac{3\pi}{4}\right)$. Using allied angles: $\cos\dfrac{3\pi}{4}=\cos\left(\pi-\dfrac{\pi}{4}\right)=-\cos\dfrac{\pi}{4}=-\dfrac{1}{\sqrt2}$, and $\sin\dfrac{3\pi}{4}=\sin\left(\pi-\dfrac{\pi}{4}\right)=\sin\dfrac{\pi}{4}=\dfrac{1}{\sqrt2}$. So $z=\sqrt2\left(-\dfrac{1}{\sqrt2}+\dfrac{1}{\sqrt2}i\right)=-1+i$. Worked Example: express (i) $3e^{5\pi i/12}\times4e^{\pi i/12}$ and (ii) $\dfrac{\sqrt2\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)}{2\left(\cos\frac{\pi}{12}+i\sin\frac{\pi}{12}\right)}$ in $a+ib$ form.

(i) $3e^{5\pi i/12}\times4e^{\pi i/12}=(3\times4)e^{i\left(\frac{5\pi}{12}+\frac{\pi}{12}\right)}=12e^{6\pi i/12}=12e^{i\pi/2}=12\left(\cos\dfrac{\pi}{2}+i\sin\dfrac{\pi}{2}\right)=12(0+i)=12i$. …