Example Problem Set 10

(A) Newton's definition of a derivative in his formulation of the calculus only captured the notion of integer-order differentiation, e.g. the first or second derivative, etc. But in scientific computing we sometimes need a notion of fractional-order derivatives, as for example in fluid mechanics.

Explain how ``Fractional Differentiation" (derivatives of non-integer order) can be given precise quantitative meaning through Fourier analysis.

Suppose that a continuous function f(x) has Fourier Transform $F(\mu)$. Outline an algorithm (as a sequence of mathematical steps, not an actual program) for computing the $1.5^{\mbox{th}}$ derivative of some function f(x)

$$\frac{d^{(1.5)}f(x)}{dx^{(1.5)}}$$

(B) For $i = \sqrt{-1}$, consider the quantity $\sqrt{i}$.

Express $\sqrt{i}$ as a complex exponential. In which quadrant of the complex plane does it lie? What is the real part of $\sqrt{i}$? What is the imaginary part of $\sqrt{i}$? What is the length (the modulus) of $\sqrt{i}$?