Alien's Mathematics




All rooms lead to white hole (PDF).
Please see the bookmarks for details

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Riemann Zeta Function
00 Summary (html)
01 Zeta Generating Functions (285KB)
02 Formulas for Riemann Zeta at natural number (162KB)
03 Formulas for Riemann Zeta at odd number (186KB)
04 Formulas for Riemann Zeta at even number (118KB)
05 Formulas for Riemann Zeta at complex number (103KB)
06 Global Definition of Riemann Zeta, and generalization of related coefficients (272KB)

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Dirichlet Beta Function
00 Summary (html)
01 Dirichlet Beta Generating Functions (138KB)
02 Formulas for Dirichlet Beta (128KB)
03 Global Definition of Dirichlet Beta, and generalized Euler Nuamber (157KB)

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Higher Calculus & Super Calculus
  Integrating sinh x with respect to x  i / π times,  we obtain {e^x - e^(-1-x) } / 2.
Differentiating log x with respect to x  1/2 times,  we obtain {log x - ψ(1/2) - γ} / sqrt(πx).
 These calculations are called Fractional Calculus.  Fractional Calculus is the calculus with a fixed lower limit  that is based on Liemann-Liouville integral.4
  In contrast,  I thought about the calculus with a variable lower limit  that was based on higher order calculus,  and I named this
Super Calculus.
By making the lower limit variable,  now the non-integer times calculus of trigonometric functions and hyperbolic functions also can be included as one case of Super Calculus.


00 Summary (html)
01 Gamma Function & Digamma Functions (147KB)
02 Multifactorial (142KB)
03 Generalized Multinomial Theorem (187KB)
04 Higher Integral (321KB)
05 Termwise Higher Integral (Trigonometric, Hyperbolic) (406KB)
06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic) (143KB)
07 Super Integral (Non-integer order Integral) (264KB)
08 Termwise Super Integral (179KB)
09 Higher Derivative (142KB)
10 Termwise Higher Derivative (Trigonometric, Hyperbolic) (225KB)
11 Termwise Higher Derivative (Inv-Trigonometric, Inv-Hyperbolic) (92KB)
12 Super Derivative (Non-integer order Derivative) (278KB)
13 Termwise Super Derivative (179KB)
14 Higher and Super Calculus of Logarithmic Integral etc. (191KB)
15 Higher and Super Calculus of Elliptic Integral (195KB)
16 Higher Integral of the product of two functions (604KB)
17 Super Integral of the product of two functions (350KB)
18 Higher Derivative of the product of two functions (247KB)
19 Super Derivative of the product of two functions (350KB)
20 Higher Calculus of the product of many functions (293KB)
21 Super Calculus of the product of many functions (217KB)
22 Higher Derivative of Composition (165KB)
23 Higher integral of Composition (429KB)
24 Sugioka's Theorem on the Series of Higher Integrals (370KB)

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A la Carte
01 Generalized Taylor's Theorem (209KB)
02 Multiple Series & Exponetial Function (203KB)
03 Higher Calculus of Binomial Identity (77KB)
04 Euler-Maclaurin Summation Formula (239KB)
05 Generalized Bernoulli Polynomials and Numbers (140KB)
06 Superellipse (Lame curve) (190KB)
07 New Formula for the Sum of Powers (103KB)
08 Taylor series and Maclaurin series (184KB)
09 Absolute Value of Gamma Function (1005KB)

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e-mail : alien@kxf.biglobe.ne.jp

        宇宙人の数学  ... Japanese version.