linear fractional transformation

线性分式变换（linear fractional transformation）是指将一个复平面上的点通过一个分式函数映射到另一个复平面上的点的变换。它是复变函数中的一个重要概念，常用于解析几何、复变函数论、拓扑学等领域。线性分式变换具有保角性、保距性等重要性质，在许多数学和物理问题中都有广泛应用。

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fractional Mellin transform

The fractional Mellin transform is a generalization of the classical Mellin transform, which is a mathematical operation that maps a function of a real variable to a function of a complex variable. The fractional Mellin transform extends this concept by allowing the order of the transform to take on fractional values.

The fractional Mellin transform is defined as:

M_{a,b}^\alpha(f(x)) = \frac{1}{\Gamma(\alpha)} \int_0^\infty x^{a-1} (\ln x)^{b-1} f(x) K_\alpha\left(\frac{x}{z}\right) \frac{dz}{z},

where a and b are real constants, f(x) is the function being transformed, and K_\alpha(x) is the modified Bessel function of the second kind.

The order of the transform, denoted by \alpha, determines the degree of smoothness of the transformed function. When \alpha is an integer, the fractional Mellin transform reduces to the classical Mellin transform. When \alpha is a non-integer, the fractional Mellin transform can be used to analyze functions that do not have integer order derivatives, such as fractal functions.

The fractional Mellin transform has applications in a variety of fields, including signal processing, image processing, and fractal analysis. It has been used to analyze the scaling properties of complex systems, to extract features from images, and to filter signals.

fractional order python

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