*3.3. Fractional Calculus*

Gamma function, also called generalized factorial, is often used in the definition and operation of fractional calculus. The integral form defined by the Gamma function is described as

$$\Gamma(\mathbf{z}) = \int\_0^\infty \mathbf{e}^{-\mathbf{t}} \mathbf{t}^{\mathbf{z}-1} \mathbf{d}t, \text{Re}(\mathbf{z}) > 0. \tag{2}$$

The limit of the definition of gamma function can be expressed as follows

$$\Gamma(\mathbf{z}) = \lim\_{\mathbf{x} \to \mathbf{s}} \frac{\mathbf{n}! \mathbf{n}^{\mathbf{z}}}{\mathbf{z}(\mathbf{z}+1) \cdots (\mathbf{z}+\mathbf{n})}. \tag{3}$$

Grünwald–Letnikov fractional derivative has been generalized from the definition of integer-order derivative. For any real number p, suppose that function f(x) has continuous derivative of m + 1 in the interval [a,t]. Then, p-order derivative for f(x) can be defined as follows:

$$\mathbf{^B\_t D\_t^P f}(\mathbf{x}) = \lim\_{\mathbf{h} \to 0} \frac{1}{\mathbf{h}^\mathbf{P}} \sum\_{\mathbf{j}=0}^{\left[ (\mathbf{t} - \mathbf{p})/\mathbf{h} \right]} (-1)^\mathbf{j} \binom{\mathbf{P}}{\mathbf{j}} \mathbf{f}(\mathbf{x} - \mathbf{j} \mathbf{h}), \tag{4}$$

where h is the step size and [(t−p)/h] represents the integer part of (t−p)/h. When p is a positive real number, Equation (4) represents p-order derivative. If p is a negative real number, Equation (4) represents p-order integral.

1-order derivative of function f(x) is defined as

$$\text{f'}\left(\mathbf{x}\right) = \lim\_{\mathbf{h}\to 0} \frac{\mathbf{f}(\mathbf{x} + \mathbf{h}) - \mathbf{f}(\mathbf{x})}{\mathbf{h}}.\tag{5}$$

2-order derivative of function f(x) is described as

$$\mathbf{f}^{\prime\prime}(\mathbf{x}) = \lim\_{\mathbf{h} \to 0} \frac{\mathbf{f}^{\prime}(\mathbf{x} + \mathbf{h}) - \mathbf{f}^{\prime}(\mathbf{x})}{\mathbf{h}} = \lim\_{\mathbf{h} \to 0} \frac{\mathbf{f}(\mathbf{x} + 2\mathbf{h}) - 2\mathbf{f}(\mathbf{x} + \mathbf{h}) + \mathbf{f}(\mathbf{x})}{\mathbf{h}^{2}}.\tag{6}$$

If the derivative order of function f(x) is raised to higher order of p, then the p-order derivative of function f(x) is expressed as

$$\mathbf{f}^{(\mathbf{p})}(\mathbf{x}) = \lim\_{\mathbf{h} \to 0} \frac{1}{\mathbf{h}^{\mathbf{P}}} \sum\_{\mathbf{m}=0}^{\mathbf{P}} (-1)^{\mathbf{m}} \binom{\mathbf{P}}{\mathbf{m}} \mathbf{f}(\mathbf{x} - \mathbf{m} \mathbf{h}).\tag{7}$$

If we use Gamma function to replace the binomial coefficient of Equation (7) and extend the derivative order to a non-integer order, we can get the Grünwald–Letnikov fractional derivative in Equation (4). Since the re-sampling interval of ASD (Analytica Spectra Devices) spectrometer was 1 nm, in Equation (4), let h = 1, and then the derivative expression of v-order derivative for function f(x) can be deduced as follows:

$$\frac{d^v \mathbf{f}(\mathbf{x})}{d\mathbf{x}^v} \approx \mathbf{f}(\mathbf{x}) + (-\mathbf{v})\mathbf{f}(\mathbf{x}-1) + \frac{(-\mathbf{v})(-\mathbf{v}+1)}{2}\mathbf{f}(\mathbf{x}-2) + \frac{(-\mathbf{v})(-\mathbf{v}+1)(-\mathbf{v}+2)}{6}\mathbf{f}(\mathbf{x}-3) + \dots + \frac{\mathbf{r}(-\mathbf{v}+1)}{n!\Gamma(-\mathbf{v}+n+1)}\mathbf{f}(\mathbf{x}-\mathbf{n}).\tag{8}$$

In particular, when v = 1, 2, it is consistent with first-order and second-order derivative formulas of spectrum, respectively. From Equation (8), we can see that fractional derivatives have global and memory characteristics.
