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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 (238KB)

07 Completed Riemann Zeta (411KB)

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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 Number (147KB)

04 Completed Dirichlet Beta (169KB) *New !*

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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 (568KB)

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 (426KB)

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 (111KB) *Updatd !*

08 Taylor series and Maclaurin series (184KB)

09 Absolute Value of Gamma Function (515KB)

10 Convergence Acceleration & Summation Method by Double Series of
Functions (471KB)

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*Update History*