Exponential (Euler's) Form (DP IB Applications & Interpretation (AI): HL): Revision Note

Exponential (Euler's) form

What is exponential (Euler) form?

• The exponential (Euler) form of a complex number $z$ is

  • $z=r{\mathrm{e}}^{\mathrm{i}\theta }$

  • where

    • $r=\left|z\right|$

    • $\theta =\arg z$

• e.g. $z=1+\sqrt{3}\mathrm{i}$ has a modulus of 2 and an argument of $\frac{\pi }{3}$

  • so $z=2{\mathrm{e}}^{\frac{\mathrm{i}\pi }{3}}$

• To convert back to Cartesian form

  • go via the modulus-argument (polar) form

  • e.g. $z=4{\mathrm{e}}^{\frac{\mathrm{i}\pi }{4}}=4\left(\cos \frac{\pi }{4}+\mathrm{isin}\frac{\pi }{4}\right)$ is $4\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)=2\sqrt{2}+2\sqrt{2}\mathrm{i}$

How do I multiply and divide complex numbers in exponential (Euler) form?

• If $z_1=r_1{\mathrm{e}}^{\mathrm{i}{\theta }_1}$and $z_2=r_2{\mathrm{e}}^{\mathrm{i}{\theta }_2}$ then 

  • $z_1{z}_2=r_1{r}_2{\mathrm{e}}^{\mathrm{i}\left({\theta }_1+{\theta }_2\right)}$

    • Multiply the moduli and add the arguments

  • $\frac{z_1}{z_2}=\frac{r_1}{r_2}{\mathrm{e}}^{\mathrm{i}\left({\theta }_1-{\theta }_2\right)}$

    • Divide the moduli and subtract the arguments

• These rules makes multiplying and dividing easier in exponential (Euler) form than in Cartesian form!

• Powers of complex numbers are also easier in exponential (Euler) form

  • You can use index laws

    • e.g. $z=r{\mathrm{e}}^{\mathrm{i}\theta }$,  $z^2=r^2{\mathrm{e}}^{2\mathrm{i}\theta }$  and  $z^n=r^n{\mathrm{e}}^{\mathrm{ni}\theta }$

What numbers do I need to know in exponential (Euler) form?

• You should know the following numbers in exponential (Euler) form

• ${\mathrm{e}}^{2\pi \mathrm{i}}=1$

  • as $\cos \left(2\pi \right)=1$ and $\sin \left(2\pi \right)=0$

  • similarly $1={\mathrm{e}}^0={\mathrm{e}}^{2\pi \mathrm{i}}={\mathrm{e}}^{4\pi \mathrm{i}}={\mathrm{e}}^{6\pi \mathrm{i}}={\mathrm{e}}^{2k\pi \mathrm{i}}$

    • for all even multiples of $\pi$

• ${\mathrm{e}}^{\pi \mathrm{i}}=-1$

  • as $\cos \left(\pi \right)=-1$ and $\sin \left(\pi \right)=0$ 

  • ${\mathrm{e}}^{\pi \mathrm{i}}=-1$ can be rearranged to ${\mathrm{e}}^{\mathrm{i}\pi }+1=0$

    • known as Euler's identity

    • often considered the most elegant relationship in Mathematics!

• ${\mathrm{e}}^{\frac{\pi }{2}\mathrm{i}}=\mathrm{i}$

  • as $\cos \left(\frac{\pi }{2}\right)=0$ and $\sin \left(\frac{\pi }{2}\right)=1$