fractional-calculus.com - information about domain
- Server Status: available
- Server IP: 174.122.150.67
- Server Response Time: 312ms
Site title
Alien's Mathematics
Keywords
fractional,calculus,super,integral,alien
Description
Higher (order) calculus, Super(Fractional) calculus, etc.
Alexa information
Alexa rank: not in database
Rank Trend
Bounce Trend
DNS Information
| Host: |
Class: |
Type: |
IP: |
TTL: |
Target: |
Other: |
| fractional-calculus.com |
IN |
NS |
|
44489 |
ns2548.hostgator.com |
|
| fractional-calculus.com |
IN |
NS |
|
44489 |
ns2547.hostgator.com |
|
Server Location
Site source
Show / hide source
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<HTML>
<HEAD>
<META http-equiv="content-type" content="text/html; charset=UTF-8">
<TITLE>Alien's Mathematics</TITLE>
<META NAME="ROBOTS" CONTENT="ALL">
<META name="description" content="Higher (order) calculus, Super(Fractional) calculus, etc.">
<META name="keywords" content="fractional,calculus,super,integral,alien's">
<META http-equiv="Content-Style-Type" content="text/css">
<META name="GENERATOR" content="IBM WebSphere Studio Homepage Builder Version 9.0.0.0 for Windows">
</HEAD>
<BODY text="#ffffff" link="#FFFF00" background="hosizora.jpg" vlink="#ffff00">
<NOSCRIPT></NOSCRIPT>
<P align="center">
<IMG src="alien.gif" width="208" height="68" border="0" alt="Alien's">
<IMG src="a_ilst020.gif" width="86" height="75" border="0" align="bottom">
<IMG src="mathematics.gif" width="271" height="72" border="0" alt="Mathematics"><BR>
<BR>
<FONT color="#00ffff" size="+1"><A href="#07 New Formula for the Sum of Powers"><IMG src="sigma.jpg" border="0" width="414" height="44"></A></FONT><BR>
<BR>
<img src="/cgi-sys/Count.cgi?df=alien.dat|display=Counter|ft=0|md=5|frgb=100;139;216|dd=C"></P>
<P align="center"><FONT color="#ffffff">All rooms lead to white hole (PDF).<BR>
Please see the bookmarks for details<BR>
</FONT></P>
<P align="center"><FONT color="#ffffff">·</FONT></P>
<P><FONT size="+2" color="#ff00ff"><FONT face="ＭＳ Ｐゴシック"><FONT size="+2" color="#ff00ff"><FONT size="+2" color="#ff00ff"><I>Riemann Zeta Function</I></FONT></FONT></FONT></FONT><BR>
<FONT face="Century"><A href="summary_riemann_zeta.htm"><B><FONT face="Century" color="#00ff00">00 Summary </FONT><FONT face="Century" color="#00ff00" size="-1">(HTML)</FONT></B></A></FONT><BR>
<FONT face="Century"><A href="zeta_generating_functions.pdf">01 Zeta Generating Functions</A> </FONT><FONT face="Century" size="-1">(769KB)</FONT><BR>
<FONT face="Century"><A href="formulas_riemann_zeta_natural.pdf">02 Formulas for Riemann Zeta at natural number</A> </FONT><FONT face="Century" size="-1">(382KB)</FONT><BR>
<FONT face="Century"><A href="formulas_riemann_zeta_odd.pdf">03 Formulas for Riemann Zeta at odd number</A> </FONT><FONT face="Century" size="-1">(395KB)</FONT><BR>
<FONT face="Century"><A href="formulas_riemann_zeta_even.pdf">04 Formulas for Riemann Zeta at even number</A> </FONT><FONT face="Century" size="-1">(362KB)</FONT><BR>
<FONT face="Century"><A href="formulas_riemann_zeta_real.pdf">05 Formulas for Riemann Zeta at real number</A> </FONT><FONT face="Century" size="-1">(267KB)<FONT size="+1"><B><I><FONT color="#ff0000"> New!</FONT></I></B></FONT></FONT><BR></P>
<P align="center"><FONT color="#ffffff">·</FONT></P>
<P><FONT size="+2" color="#ff00ff"><FONT face="ＭＳ Ｐゴシック"><FONT size="+2" color="#ff00ff"><I>Dirichlet Beta Function<BR>
</I><FONT face="ＭＳ ゴシック"><FONT face="Century"><A href="summary_dirichlet_beta.htm"><B><FONT color="#00ff00"><FONT size="-1"><FONT size="+0">00 Summary </FONT></FONT></FONT><FONT color="#00ff00" size="-1">(HTML)</FONT></B></A></FONT></FONT><BR>
</FONT><FONT size="+1" color="#ff00ff"><FONT face="Century"><A href="dirichlet_beta_generating_func.pdf">01 Dirichlet Beta Generating Functions</A> </FONT><FONT face="Century" size="-1"><FONT size="+1" color="#ffffff"><FONT face="Century"> </FONT><FONT face="Century" size="-1">(385KB)</FONT><BR>
</FONT></FONT></FONT></FONT></FONT><FONT face="Century" size="-1"><FONT face="ＭＳ ゴシック"><FONT face="Century" size="-1"><FONT face="ＭＳ ゴシック"><FONT face="Century" size="-1" ><FONT face="ＭＳ ゴシック"><FONT face="Century" size="-1" ><FONT face="Century"><A href="formulas_dirichlet_beta.pdf"><FONT size="+0">02 Formulas for Dirichlet Beta</FONT></A> </FONT>(446KB)<BR>
</FONT></FONT></FONT></FONT></FONT></FONT></FONT></P>
<P align="center"><FONT color="#ffffff">·</FONT></P>
<P><FONT size="+2" color="#ff00ff"><FONT size="+2" color="#ff00ff" face="ＭＳ Ｐゴシック">
<FONT size="+2" color="#ff00ff" face="ＭＳ Ｐゴシック"><FONT size="+2" color="#ff00ff" face="ＭＳ Ｐゴシック"><I>Higher Calculus & Super Calculus</I></FONT></FONT></FONT><BR>
</FONT> <FONT size="+1" color="#ffffff">Integrating sinh x with respect to x i / π times, we
obtain {e^x - e^(-1-x) } / 2.<BR>
Differentiating log x with respect to x 1/2 times, we
obtain {log x - ψ(1/2) - γ} / sqrt(πx).<BR>
These calculations are called <I>Fractional Calculus</I>. Fractional Calculus is the calculus<BR>
with a fixed lower limit that is based on Liemann-Liouville integral.<BR>
In contrast, I thought about the calculus with a variable
lower limit that was based on<BR>
higher order calculus, and I named this <I>Super Calculus</I>.<BR>
By making the lower limit variable, now the non-integer times
calculus of trigonometric<BR>
functions and hyperbolic functions also can be included as one case of <I>Super Calculus</I>.<BR></FONT><FONT size="+2" color="#ff00ff"><BR></FONT>
<FONT face="Century"><A href="summary_calculus.htm"><B><FONT face="Century" color="#00ff00">00 Summary </FONT><FONT face="Century" color="#00ff00" size="-1">(HTML)</FONT></B></A></FONT><BR>
<FONT face="Century"><A href="gamma_digamma.pdf">01 Gamma Function & Digamma Function</A> </FONT><FONT face="Century" size="-1">(315KB)</FONT><BR>
<FONT face="Century"><A href="multifactorial.pdf">02 Multifactorial</A> </FONT><FONT face="Century" size="-1">(290KB)</FONT><BR>
<FONT face="Century"><A href="multinomial_theorem.pdf">03 Generalized Multinomial Theorem</A> </FONT><FONT face="Century" size="-1">(465KB)</FONT><BR>
<FONT face="Century"><A href="higher_integral.pdf">04 Higher Integral</A> </FONT><FONT face="Century" size="-1">(822KB)</FONT><BR>
<FONT face="Century"><A href="termwise_higher_integral_1.pdf">05 Termwise Higher Integral (Trigonometric, Hyperbolic)</A> </FONT><FONT face="Century" size="-1">(1000KB)</FONT><BR>
<FONT face="Century"><A href="termwise_higher_integral_2.pdf">06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic)</A> </FONT><FONT face="Century" size="-1">(489KB)</FONT><BR>
<FONT face="Century"><A href="super_integral.pdf">07 Super Integral (Non-integer order Integral)</A> </FONT><FONT face="Century" size="-1">(858KB)</FONT><BR>
<FONT face="Century"><A href="termwise_super_integral.pdf">08 Termwise Super Integral</A> </FONT><FONT face="Century" size="-1">(406KB)</FONT><BR>
<FONT face="Century"><A href="higher_derivative.pdf">09 Higher Derivative</A> </FONT><FONT size="-1" face="Century">(515KB)</FONT><BR>
<FONT face="Century"><A href="termwise_higher_derivative_1.pdf">10 Termwise Higher Derivative (Trigonometric, Hyperbolic)</A> </FONT><FONT face="Century" size="-1">(861KB)</FONT><BR>
<FONT face="Century"><A href="termwise_higher_derivative_2.pdf">11 Termwise Higher Derivative (Inv-Trigonometric, Inv-Hyperbolic)</A> </FONT><FONT face="Century" size="-1">(199KB)</FONT><BR>
<FONT face="Century"><A href="super_derivative.pdf">12 Super Derivative (Non-integer order Derivative)</A> </FONT><FONT size="-1" face="Century">(928KB)</FONT><BR>
<FONT face="Century"><A href="termwise_super_derivative.pdf">13 Termwise Super Derivative</A> </FONT><FONT size="-1" face="Century">(454KB)</FONT><BR>
<FONT face="Century"><A href="calculus_logarithmic_integral.pdf">14 Higher and Super Calculus of Logarithmic Integral etc.</A> </FONT><FONT size="-1" face="Century">(723KB)</FONT><BR>
<FONT face="Century"><A href="calculus_elliptic_integral.pdf">15 Higher and Super Calculus of Elliptic Integral</A> </FONT><FONT size="-1" face="Century">(683KB)</FONT><BR>
<FONT face="Century"><A href="higher_integral_product_two.pdf">16 Higher Integral of the product of two functions</A> </FONT><FONT size="-1" face="Century">(1,501KB)</FONT><BR>
<FONT face="Century"><A href="super_integral_product_two.pdf">17 Super Integral of the product of two functions</A> </FONT><FONT size="-1" face="Century">(959KB)</FONT><BR>
<FONT face="Century"><A href="higher_derivative_product_two.pdf">18 Higher Derivative of the product of two functions</A> </FONT><FONT size="-1" face="Century">(713KB)</FONT><BR>
<FONT face="Century"><A href="super_derivative_product_two.pdf">19 Super Derivative of the product of two functions</A> </FONT><FONT size="-1" face="Century">(948KB)</FONT><BR>
<FONT face="Century"><A href="higher_calculus_product_many.pdf">20 Higher Calculus of the product of many functions</A> </FONT><FONT size="-1" face="Century">(444KB)</FONT><BR>
<FONT face="Century"><A href="super_calculus_product_many.pdf">21 Super Calculus of the product of many functions</A> </FONT><FONT size="-1" face="Century">(363KB)</FONT><BR>
<FONT face="Century"><A href="higher_derivative_composition.pdf">22 Higher Derivative of Composition</A> </FONT><FONT size="-1" face="Century">(303KB)</FONT><BR>
<FONT face="Century"><A href="higher_integral_composition.pdf">23 Higher integral of Composition</A> </FONT><FONT size="-1" face="Century">(1091KB)</FONT></P>
<P align="center"><FONT color="#ffffff">·</FONT></P>
<P align="left"><FONT size="+2" color="#ff00ff"><FONT face="ＭＳ Ｐゴシック"><FONT size="+2" color="#ff00ff"><FONT size="+2" color="#ff00ff"><I>A la Carte</I></FONT></FONT></FONT></FONT><BR>
<FONT face="Century"><A href="generalized_taylor's_theorem.pdf">01 Generalized Taylor's Theorem</A> </FONT> <FONT face="Century" size="-1">(315KB)</FONT><BR>
<FONT face="Century"><A href="multiple_series_exponetial_func.pdf">02 Multiple Series & Exponetial Function</A> </FONT><FONT face="Century" size="-1">(670KB)</FONT><BR>
<FONT face="Century"><A href="higher_calculus_binomial_identity.pdf">03 Higher Calculus of Binomial Identity</A> </FONT><FONT face="Century" size="-1">(231KB)<FONT size="+1"><B><I><FONT color="#ff0000"></FONT></I></B></FONT></FONT><BR>
<FONT face="Century"><A href="euler_maclaurin_sum_formula.pdf">04 Euler-Maclaurin Summation Formula</A> </FONT><FONT face="Century" size="-1">(585KB)</FONT><BR>
<FONT face="Century"><A href="generalized_bernoulli_number.pdf">05 Generalized Bernoulli Polynomials and Numbers</A> </FONT><FONT face="Century" size="-1">(378KB)</FONT><BR>
<FONT face="Century"><A href="super_ellipse.pdf">06 Superellipse (Lame curve)</A> </FONT><FONT face="Century" size="-1">(617KB)</FONT><BR>
<FONT face="Century"><A href="new_formula_sum_of_powers.pdf" name="07 New Formula for the Sum of Powers">07 New Formula for the Sum of Powers</A> </FONT><FONT face="Century" size="-1">(372KB)<FONT size="+1"><B><I><FONT color="#ff0000"> New!</FONT></I></B></FONT></FONT></P>
<P align="center"><FONT color="#ffffff">·</FONT></P>
<P align="center"><B><I><FONT color="#00ffff" size="+1">Update History</FONT></I></B></P>
<FORM>
<CENTER><TEXTAREA rows="8" cols="94" style="color:#FFFFFF; font-size:9pt; background-color:000000">
2012.03.16 Added "2.4 Formulas for Beta at Real Number" to "02 Formulas for Dirichlet Beta"
2012.03.14 Uploaded "05 Formulas for Riemann Zeta at real number"
2012.02.25 Added "2.3 Formulas for Beta at Odd Number" to "02 Formulas for Dirichlet Beta"
2012.02.24 Uploaded "04 Formulas for Riemann Zeta at even number"
2012.02.16 Renewed Dirichlet Beta Function.
The formulas were generalized so that those hold for 0<x<=PI/2.
2012.02.05 Renewed Riemann Zeta Function.
The formulas were generalized so that those hold for 0<x<=PI.
2012.01.06 Renewed "09 Higher Derivative". Added higher derivatives of 14 functions.
2011.12.24 Uploaded "07 New Formula for the Sum of Powers" to A la Carte
2011.11.05 Uploaded "06 Superellipse (Lame curve)" to A la Carte
2011.09.27 Corrected the description in "15.2 Arc length of an ellipse"
2011.09.07 Corrected Example of Formula 4.6.1 in "04 Higher Integral"
2011.08.25 Updated "04 Euler-Maclaurin Summation Formula" (Added 4.6)
2011.08.25 Uploaded "05 Generalized Bernoulli Polynomials and Numbers" to A la Carte
2011.07.25 Uploaded "04 Euler-Maclaurin Summation Formula" to A la Carte
2011.07.11 Uploaded "03 Higher Calculus of Binomial Identity" to A la Carte
2011.07.09 Uploaded "02 Multiple Series & Exponential Function" to A la Carte
2011.07.05 Corrected the proof of Formula 1.1.4 (2) in A la carte
2011.06.15 Uploaded "01 Generalized Taylor's Theorem" to A la Carte newly established
2011.05.10 Added Summary (HTML) to Dirichlet Beta Function
2011.05.09 Added Summary (HTML) to Riemann Zeta Function
2011.05.06 Simplified "Superfluous Formula for Zeta(n)" in "02 Formulas for Riemann Zeta"
2011.04.16 Added Summary (HTML) to Higher Calculus & Super Calculus
2011.03.26 Uploaded "01 Dirichlet Beta Generating Functions" and
"02 Formulas for Dirichlet Beta"
2011.03.23 Added 3 formulas for Zeta at natural numbers to "01 Zeta Generating Functions"
2011.03.07 Uploaded "02 Formulas for Riemann Zeta"
2011.03.02 Uploaded "01 Zeta Generating Functions"
2011.02.15 Simplified the description of "06 Termwise Higher Integral(Inv-Trigono. etc.)"
2011.02.07 Added collateral higher integrals of inv-trigonometric func. and
inv-hyperbolic func. to "04 Higher Integral"
2011.02.04 Added higher integrals of inv-trigonometric func. and inv-hyperbolic func.
to "04 Higher Integral"
2011.01.15 Uploaded "23 Higher Integral of Composition"
2011.01.07 Uploaded "22 Higher Derivative of Composition"
2010.12.29 Uploaded "21 Super Calculus of the product of many functions".
2010.12.21 Simplified the proofs of Formula20.2.3, Formula20.2.4,and etc.
2010.12.18 Restructured Formula20.2.3 and Formula20.2.4.
2010.12.14 Corrected Formula20.2.3 and Formula20.2.4.
2010.12.10 Corrected the condition of Theorem17.1.2, Theorem17.1.2' and Formula17.2.0.
2010.12.09 Corrected the condition of Theorem20.2.1 and Theorem20.2.2.
2010.12.01 Uploaded "20 Higher Calculus of the product of many functions".
2010.11.19 Uploaded the following chapters.
15 Higher and Super Calculus of Elliptic Integral
16 Higher Integral of the product of two functions
17 Super Integral of the product of two functions
18 Higher Derivative of the product of two functions
19 Super Derivative of the product of two functions
2010.11.18 Uploaded the following chapters.
08 Termwise Super Integral
09 Higher Derivative
10 Termwise Higher Derivative (Trigonometric, Hyperbolic)
11 Termwise Higher Derivative (Inv-Trigonometric, Inv-Hyperbolic)
12 Super Derivative (Non-integer order Derivative)
13 Termwise Super Derivative
14 Higher and Super Calculus of Logarithmic Integral etc.
2010.11.16 This homepage has moved from Infoseek.
Uploaded the following chapters.
01 Gamma Function & Digamma Function
02 Multifactorial
03 Generalized Multinomial Theorem
04 Higher Integral
05 Termwise Higher Integral (Trigonometric, Hyperbolic)
06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic)
07 Super Integral (Non-integer order Integral)</TEXTAREA></CENTER></FORM>
<HR noshade>
<P>e-mail : alien@fractional-calculus.com</P>
<P align="center"> <A href="http://www.geocities.jp/uchu_tako/newpage2.html">
<FONT color="#00ff00"><IMG src="logo1.gif" width="172" height="22" border="0" alt="宇宙人の数学"></FONT></A> ... Japanese version.</P>
</BODY>
</HTML>