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Mathematics of the DFT
    Proof of Euler's Identity
       Back to Mth Roots

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Back to Mth Roots

As mentioned in §3.4, there are $ M$ different numbers $ r$ which satisfy $ r^M=a$ when $ M$ is a positive integer. That is, the $ M$th root of $ a$, which is written as $ a^{1/M}$, is not unique--there are $ M$ of them. How do we find them all? The answer is to consider complex numbers in polar form. By Euler's Identity, which we just proved, any number, real or complex, can be written in polar form as

$\displaystyle z = r e^{j\theta} $

where $ r\geq 0$ and $ \theta\in[-\pi,\pi)$ are real numbers. Since, by Euler's identity, $ e^{j2\pi k}=\cos(2\pi k)+j\sin(2\pi k)=1$ for every integer $ k$, we also have

$\displaystyle z = r e^{j\theta} e^{j2\pi k}. $

Taking the $ M$th root gives

$\displaystyle z^{\frac{1}{M}} = \left(r e^{j\theta} e^{j2\pi k}\right)^{\frac{... ...rac{k}{M}} = r^{\frac{1}{M}} e^{j\frac{\theta+2\pi k}{M}}, \quad k\in {\bf Z}. $

There are $ M$ different results obtainable using different values of $ k$, e.g., $ k=0,1,2,\dots,M-1$. When $ k=M$, we get the same thing as when $ k=0$. When $ k=M+1$, we get the same thing as when $ k=1$, and so on, so there are only $ M$ distinct cases. Thus, we may define the $ k$th $ M$th-root of $ z=r e^{j\theta}$ as

$\displaystyle r^{\frac{1}{M}} e^{j\frac{\theta+2\pi k}{M}}, \quad k=0,1,2,\dots,M-1. $

These are the $ M$ $ M$th-roots of the complex number $ z=r e^{j\theta}$.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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