## 数学：K函数

In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function.

Formally, the K-function is defined as

It can also be given in closed form as

where ζ'(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and

Another expression using polygamma function is[1]

where A is Glaisher constant.

The K-function is closely related to the gamma function and the Barnes G-function; for natural numbers n, we have

More prosaically, one may write

The first values are

1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, … ((sequence A002109 in the OEIS)).

## 物理学：引力的修改理论图解

While Einstein’s general relativity has been well tested by various observations, it is possible that at the cosmological scale it might require some modification or extension, which could, in turn, explain dark matter or dark energy. There exist a plethora of modified theories of gravity, from simple straightforward generalizations of the Hilbert-Einstein action (e.g. f(R) gravity), to utilizing other geometric structures (e.g. f(T) gravity that models gravity as spacetime torsion instead of curvature), it is important to constrain these theories from both observations and theoretical grounds (a good theory should be mathematically self-consistent and free of pathologies). [Figure source: Tessa Baker]

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## 物理学：如何理解黑洞的图像

4月10日晚（今晚），第一幅真实的黑洞图像就会被公布。这个视频是最近发布的（发布于昨天），讲解了如何理解黑洞的图像。