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Update History

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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 Riemann-Liouville integral.
  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 (pdf)
01 Gamma Function & Digamma Functions (226KB)
02 Multifactorial (142KB)
03 Generalized Multinomial Theorem (309KB)
04 Higher Integral (321KB)
05 Termwise Higher Integral (Trigonometric, Hyperbolic) (488KB)
06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic) (143KB)
07 Super Integral (Non-integer order Integral) (339KB)
08 Termwise Super Integral (179KB)
09 Higher Derivative (142KB)
10 Termwise Higher Derivative (Trigonometric, Hyperbolic) (225KB)
11 Termwise Higher Derivative (Inv-Trigonometric, Inv-Hyperbolic) (167KB)
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 (584KB)
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 (218KB)
22 Higher Derivative of Composition (213KB)
23 Higher Integral of Composition (426KB)
24 Sugioka's Theorem on the Series of Higher Integrals (503KB)
25 Series of Higher Integral with Geometric Coefficients (469KB)
26 Higher and Super Calculus of Zeta Function etc (884KB)
27 Zeros of Super Derivative of Riemann Zeta (404KB)
     Animation: Zero of ζ' (z) ← Zero of ζ(z)
28 Zeros of Super Integral of Riemann Zeta (343KB)
     Animation: Zeros of ∫ ζ(z) dz ← Zeros of ζ(z)
29 Higher Derivative of Complex Function (200KB)
30 Higher and Super Definite Integrals of Linear Composite Functions (207KB) New !

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Riemann Zeta Function
 After 17 years of research, I finally arrived at a proof of the Riemann hypothesis.

00 Summary (pdf)
01 Zeta Generating Functions (365KB)
02 Formulas for Riemann Zeta at natural number(162KB)
03 Formulas for Riemann Zeta at odd number (185KB)
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 (218KB))
07 Completed Riemann Zeta (403KB)
08 Factorization of Completed Riemann Zeta (299KB))
09 Maclaurin Series of Completed Riemann Zeta (369KB)
10 Vieta's Formulas on Completed Riemann Zeta (353KB)
11 Zeros on the Critical Line of Riemann Zeta (151KB)
12 Zeros of Riemann Zeta and System of Infinite Degree Equations (289KB)
Graphical Proof of the Riemann Hypthesis (1878KB)
Analytical Proof of the Riemann Hypthesis (350KB)

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Dirichlet Beta Function
 After 17 years of research, I finally arrived at a proof of the Riemann hypothesis.

00 Summary (pdf)
01 Dirichlet Beta Generating Functions(138KB)
02 Formulas for Dirichlet Beta (127KB)
03 Global Definition of Dirichlet Beta, and generalized Euler Number (147KB)
04 Completed Dirichlet Beta (201KB)
05 Factorization of Completed Dirichlet Beta (224KB)
06 Zeros on the Critical Line of Dirichlet Beta (142KB)
07 Absolute Value of Dirichlet Beta Function (569KB)
Graphical Proof of the Riemann Hypthesis for the Dirichlet Beta Function (1886KB)
Analytical Proof of the Riemann Hypthesis for the Dirichlet Beta Function (345KB)

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Dirichlet Series
00 List of Theorems and Formulas (pdf)
01 General Dirichlet Series & Power Series (655KB)
02 Dirichlet Series & Logarithmic Power Series (823KB)
03 Complementary Series of Dirichlet Series (1171KB)
04 Absolute Value of Dirichlet Eta Function (777KB)
05 Split of Dirichlet Series (334KB)
06 Reflection Split of Dirichlet Series (292KB)
07 Zeros of p-series (446KB)
08 Series Expansion of Dirichlet Eta Function (879KB)
09 Power Series of Riemann Zeta etc by Real & Imaginary Parts (616KB)
10 Dirichlet Series & Taylor Series (537KB)
11 Zeros of Dirichlet Eta and System of Transcendental Equations (461KB)
12 Dirichlet Eta type Cosine Series (930KB)
13 Dirichlet Eta type Sine Series (862KB)
14 Zeros and Phases in Dirichlet Series (208KB)

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Infinite-degree Equation
00 List of Theorems and Formulas (pdf)
01 Power of Infinite Series.pdf (320KB)
02 Infinite-degree Equation with Integers as Roots (445KB)
03 Vieta's Formulas in Infinite-degree Equation (486KB)
04 Sum of series equivalent to the Riemann hypothesis (340KB)

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A la Carte
00 List of Theorems and Formulas (pdf)
01 Generalized Taylor's Theorem (281KB)
02 Multiple Series & Exponetial Function (195KB)
03 Higher Calculus of Binomial Identity (77KB)
04 Euler-Maclaurin Summation Formula (410KB)
05 Generalized Bernoulli Polynomials and Numbers (140KB)
06 Superellipse (Lame curve) (253KB)
07 New Formula for the Sum of Powers (194KB) Renewal !
08 Taylor series and Maclaurin series (296KB)
09 Absolute Value of Gamma Function (515KB)
10 Convergence Acceleration & Summation Method by Double Series of Functions (471KB)
11 Series Expansion of Reciprocal Gamma Function (326KB)
12 Series Expansion of Gamma Function & the Reciprocal (472KB)
13 Convergence Acceleration of Multiple Series (235KB)
14 Taylor Expansion by Real Part & Imaginary Part (1073KB)
15 Taylor Series of Elementary Functions by Real & Imaginary Parts (539KB)
16 Split of Power Series (332KB)
17 Expression of Polynomial with Real Coefficients by Real & Imaginaryparts (339KB)
18 Power Series with the signs of the terms inverted equidistantly (205KB)
19 Composition of Trigonometric Functions (526KB)
20 Series Expansion by Real & Imaginary Parts of Gamma Function (216KB)
21 Taylor Expansion by Real & Imaginary Parts around a Complex Number (561KB)

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