Solution Notes -- Example Problem Set 10

(A) By generalization of the Differentiation Theorem of Fourier Analysis, if a function f(x) has Fourier Transform $F(\mu)$, then $f^{(\beta)}(x)$, the $\beta^{th}$ derivative of f(x) (where $\beta$is not necessarily an integer) has Fourier Transform: $(i\mu)^{\beta}F(\mu)$.

The inverse Fourier Transform of this expression then gives precise quantitative meaning to Fractional Differentiation.

Algorithm:

1.

Go into the Fourier domain by taking the Fourier Transform of f(x):

$$f(x) \stackrel{FT}{\Longrightarrow} F(\mu)$$

2.

Multiply $F(\mu)$ by the desired $\beta$ power of $(i\mu)$, in this case 1.5:

$$F(\mu) \stackrel{\beta}{\Longrightarrow} (i\mu)^{1.5}F(\mu)$$

which in this case amounts to a soft high-pass filtering plus a phase rotation.

3.

Compute the inverse Fourier Transform of the result to obtain the desired new function:

$$(i\mu)^{1.5}F(\mu) \, \, \stackrel{FT^{-1}}{\Longrightarrow} \, \, \frac{d^{(1.5)}f(x)}{dx^{(1.5)}}$$

(B) Complex exponential form: $\sqrt{i} = e^{i \pi /4}$. (Negative root $\sqrt{i} = e^{i 5\pi /4}$ okay too.) The positive root lies in the first quadrant (and the negative root in the third). The real part of $\sqrt{i}$ is $1/\sqrt{2}$ (or - $1/\sqrt{2}$ for the negative root). The imaginary part of $\sqrt{i}$ is also $1/\sqrt{2}$ (or - $1/\sqrt{2}$ for the negative root). The length of $\sqrt{i}$ is 1.