Summary
01 Gamma Function & Digamma Function
Although the factorial n! and the harmonic number math0001.pngare usuallydefined for a natural
number, if a gamma function and a digamma function are used, these can be defined for the real number p. That is,
math0002.png
The former is convenient to express the coefficient of the higher order primitive or derivative of a power function, and the
latter is indispensable to non-integer order calculus of the logarithmic function.
Although some formulas about these functions are described here, the following two formulas proved in Section 3 are
especially important. That is, when math0003.png ,
math0004.png
These show that the ratios of the singular points of math0005.pngor math0006.pngreduce to integers or its reciprocals. The former is
necessary to express the higher order derivative of fractional functions, and the latter is required for the super calculus
of a logarithmic function.
02 Multifactorial
The relational expression of multifactorial and the gamma function is shown here.
For instance, in case of double factorial,
math0007.png
math0008.png
These are used to express the half-integration of a integer-power function later.
Moreover, we obtain the following Maclaurin expansions as by-products.
math0009.png
math0010.png
03 Generalized Multinomial Theorem
First, the binomial theorem and a generalized binomial theorem are mentioned. The Leibniz rule and the Leibniz rule
about super-differentiation are expressed just like these later.
Next, multinomial thorem and generalized multinomial thorem are shown as follows.
math0011.png
math0012.png
Higher order calculus of the product of many functions and super order calculus of the product of many functions are
expressed just like these later.
What should be paid attention here are the following property of generalized binomial coefficients.
math0013.png
That is, once generalized binomial coefficients were used, the upper limit of math0014.png should be math0015.png. Therefore, when n is
a natural number and p is a real number, the following holds in most cases.
math0016.png
When the original coefficient is not binomial coefficient e.g. 1 ,
math0017.png
Although math0018.pngis difficult, math0019.pngis satisfactory. What enables super calculus in this text
is just this property of the generalized binomial coefficient. Newton is great!
04 Higher Integral
(1) Definitions and Notations
The 1st order primitive function of f(x) is usually denoted F(x). However, such a notation is unsuitable for the
description of the 2nd or more order primitive functions. Then, math0020.png denote the each order
primitive functions of f(x) in this text. Here, for example, when f(x)=sin x, math0021.pngmight mean -cos x or
might mean -cos x+ c. Which it means follows the definition at that time.
Furthermore, each order integrals of f(x) are denoted as follows.
math0022.png
And these are called higher integral with variable lower limits . On the other hand,
math0023.png
are called higher integral with a fixed lower limit.
(2) Fundamental Theorem of Higher Integral
There is Fundamental Theorem of Calculus for the 1st order integral. The same theorem holds for the higher order
integral.
Theorem 4.1.3
Let math0024.png be continuous functions defined on a closed interval I and math0025.png be the arbitrary
primitive function of math0026.png. Then the following expression holds for math0027.png.
math0028.png
Especially, when math0029.png ,
math0030.png
(3) Lineal and Collateral
We call Constant-of-integration Polynomial the 2nd term of the right sides of these. And when Constant-of-integration
Polynomial is 0, we call the left side Lineal Higher Integral and we call math0031.pngLineal Higher Primitive Function.
Oppositely, when Constant-of-integration Polynomial is not 0, we call the left side Collateral Higher Integral and we
call the right side Collateral Higher Primitive Function.
For example,
math0032.png
math0033.png
Furthermore, from Theorem 4.1.3, we see that math0034.png must be all zeros of math0035.png in order for the
higher integral of f(x) to be lineal.
(4) Higher Integral and Reimann-Liouville Integral
The higher integral with a fixed lower limit reduce to the 1st order integral which called Reimann-Liouville Integral.
Theorem 4.2.1 ( Cauchy formula for repeated integration )
When f(x) is continuously integrable function and math0036.pngdenotes a gamma function ,
math0037.png
Reimann-Liouville Integral of the right side is important. However, in the higher integral, since the left side itself has
an operation functions, the right side is not indispensable.
By the way, replacing the left side in Theorem 4.1.3 for Reimann-Liouville Integral and shifting the index by -n,
we obtain the following.
math0038.png
This is the Taylor expansion of f(x) around a. And Reimann-Liouville Integral of math0039.pngis the remainder term called
Bernoulli form
(5) Higher Integrals of Elementary Functions.
When m is a natural number, the 2nd order integral of math0040.png becomes as follows.
math0041.png
Then, when math0042.pngis a positive number, it is as follows.
math0043.png
Here, math0044.pngcan be expressed by gamma function math0045.png. Thus
math0046.png
By such an easy calculation, we obtain the following expressions for elementary functions. Where, math0047.png denote the
ceiling function and the floor function respectively.
Higher Integrals of Power Function etc.
math0048.png
math0049.png
math0050.png
math0051.png
math0052.png
math0053.png
math0054.png
math0055.png
Higher Integrals of Inverse Trigonometric Functions
math0056.png
math0057.png
math0058.png
math0059.png
math0060.png
math0061.png
math0062.png
math0063.png
Where, math0064.png
math0065.png
math0066.png
Where, math0067.png
math0068.png
math0069.png
math0070.png
math0071.png
Where, math0072.png are all complex numbers.
Higher Integrals of Inverse Hyperbolic Functions
math0073.png
math0074.png
math0075.png
math0076.png
math0077.png
math0078.png
math0079.png
Where, math0080.png
math0081.png
math0082.png
math0083.png
math0084.png
Where, math0085.png, math0086.pngare complex numbers.
math0087.png
math0088.png
Where, math0089.png
(6) Termwise Higher Integral and Taylor series of higher primitive function
Theorem 4.6.1
Let math0090.png be continuous functions defined on [a,b] and math0091.pngbe the arbitrary primitive function
of math0092.png. At this time, if f(x) can be expanded to the Taylor series around a, the following expressions hold for
math0093.png.
math0094.png
This expression shows that the Taylor series of math0095.pngconsists of the constant-of-integration polynomial and the
termwise higher integral of f(x). The following can be said from this.
(1) A termwise higher integral with a fixed lower limit is collateral generally.
(2) It is the following case that a termwise higher integral with a fixed lower limit is lineal.
math0096.png
For example,
math0097.png is collateral higher integral.
math0098.png is lineal higher integral.
Next, the following is obtained from the Taylor series of the higher integral of log x.
math0099.png
math0100.png
math0101.png
math0102.png
Example
math0103.png
math0104.png
05 Termwise Higher Integral (Trigonometric, Hyperbolic)
In this chapter, for the function which second or more order integral cannot be expressed with the elementary functions
among trigonometric functions and hyperbolic functions, we integrate the series expansion of these function termwise
and obtain the following expressions. Where, math0105.png denote the ceiling function and the floor function respectively. And
Bernoulli Numbers and Euler Numbers are as follows.
math0106.png
math0107.png
(1) Taylor Series
math0108.png
math0109.png
math0110.png
(2) Fourier Series
When math0111.png,
math0112.png
math0113.png
math0114.png
math0115.png
math0116.png
math0117.png
math0118.png
(3) Riemann Odd Zeta & Dirichlet Odd Eta
Comparing Taylor series and Fourier series, we obtain Riemann Odd Zeta and Dirichlet Odd Eta. For example,
math0119.png
math0120.png
math0121.png
math0122.png
math0123.png
math0124.png
math0125.png
06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic)
Here, we integrate the series of an inverse trigonometric function or an inverse hyperbolic function term by term.
Then, we obtain formulas simpler than what were obtained in "04 HIgher Integral". Moreover, both are compared and
we obtain various by-products.
(1) Taylor Series
math0126.png
math0127.png
math0128.pngcollateral
math0129.pngcollateral
math0130.png
math0131.pngcollateral
math0132.png
0< x<1collateral
math0133.png
0< x<1collateral
(2) By-products
math0134.png
math0135.png
math0136.png
math0137.png
math0138.png
math0139.png
math0140.png
math0141.png
math0142.png
Example
math0143.png
math0144.png
math0145.png
math0146.png
07 Super Integral (Non-integer order Integral)
It is math0147.png which extended the domain of index j ofmath0148.png to the real number interval [0,p] from the natural
number interval [1,m]. And it is math0149.png which extended the domain of index k of math0150.pngto the real number
interval [0,q] from the natural number interval [1,n]. It is called analytic continuation to extend the domain
generaly. Although usually analytic continuation is used for extending the domain of a function, it can be used also
for extending the domain of the index of a operator. Here, extending the domain of the index of the integration operator,
we obtain Super Integral (Non-integer order Integral ).
(1) Definitions and Notations
math0151.pngdenotes the non-integer order primitive function of f(x). And we call this SuperPrimitive Function of
f(x). Since there is a super primitive function innumerably, which math0152.pngmeans follows the definition at that time.
We call it Super Integral to integrate a function f with respect to an independent variable x from a(0) to a(p)
continuously. And it is described as follows.
math0153.png
And
when math0154.png, we call it super integral with a fixed lower limit,
when math0155.png, we call it super integral with variable lower limits.
(2) Fundamental Theorem of Super Integral
Let math0156.pngbe an continuous function on the closed interval I and be arbitrary the r-th order primitive
function of f. And let a(r) be a continuous function on the closed interval [0,p].
Then the following expression holds for math0157.png.
math0158.png
Especially, when math0159.png,
math0160.png
(3) Lineal and Collateral
We call Constant-of-integration Function the 2nd term of the right sides of these.
And when Constant-of-integration Function is 0, we call the left side Lineal Super Integral . and we call math0161.png
Lineal Super Primitive Function.
Oppositely, when Constant-of-integration Function is not 0, we call the left side Collateral Super Integral and we call
the right side Collateral Super Primitive Function.
For example,
math0162.png
math0163.png
(4) Super Integral and Reimann-Liouville Integral
The super integral with a fixed lower limit reduce to the 1st order integral which called Reimann-Liouville Integral.
math0164.png
This is what extended the parameter n to the real number in Cauchy formula for repeated integration. Since the left
side has lost the operating function, Reimann-Liouville Integral of the right side is very important. All the super integral
with a fixed lower limit can be verified numerically by this. On the other hand, since the super integral with variable
lower limits cannot apply Reimann-Liouville Integral, the verification is vary difficult.
(5) Fractional Integral & Super Integral
In traditional Fractional Integral, the super primitive function is drawn from Riemann-Liouville Integral.
For example, in the case of math0165.png, it is as follws.
Let
math0166.png
Then
math0167.png
We find out that this is a convolution math0168.png. Then we take the Laplace transform of math0169.png
math0170.png
math0171.png
math0172.png
math0173.png
Finally, taking the inverse Laplace transform, we obtain math0174.png.
Because the technique of Fractional Integral is difficult like this, it is more difficult to obtain the super primitive
function of log x by this technique.
Above all, the problem is that Fractional Integral cannot treat the lineal non-integer order integral such as sin x.
Because Riemann-Liouville Integral cannot be applied to the integral with variable lower limets.
On the other hand, in Super Integral that I advocates, first of all, we obtain the following higher integral.
math0175.png
And replacing the index of the integration operator n with a real number p, we obtain the following very easily.
math0176.png
Furthermore, performing the higher integral with variable lower limits and replacing the index of the integration operator
with a real number, we can obtain the super integral such as sin x easily.
(6) Super Integrals of Elementary Functions.
In this way, the following super integrals obtined from " 04 Higher Integral " ,
math0177.png
math0178.png
math0179.png
math0180.png
math0181.png
math0182.png
math0183.png
math0184.png
math0185.png
math0186.png
math0187.png
math0188.png
math0189.png
math0190.png
math0191.png
math0192.png
math0193.png
(7) By-products
math0194.png
math0195.png
Where, B(x,y) is the beta function.
08 Termwise Super Integral
The following termwise super integrals are obtained from " 05 Termwise Higher Integral (Trigonometric,Hyperbolic) "
and " 06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic) ".
Where,math0196.png denote the ceiling function and the floor function respectively. And Bernoulli Numbers and Euler Numbers
are as follows.
math0197.png
math0198.png
math0199.png
math0200.png
math0201.pngcollateral
math0202.pngcollateral
math0203.png
math0204.pngcollateral
math0205.png
math0206.png
math0207.png
math0208.png
math0209.pngcollateral
math0210.pngcollateral
math0211.png
math0212.pngcollateral
math0213.png
0<x<1collateral
math0214.png
0<x<1collateral
09 Higher Derivative
(1) Definitions and Notations
When math0215.pngdenotes the derivative function of math0216.pngfor math0217.png, we call math0218.pngHigher Derivative
of f(x)..
Moreover, we call it Higher Differentiation to differentiate a function f with respect to an independent variable x
repeatedly. And it is described as follows.
math0219.png
(2) Fundamental Theorem of Higher Differentiation
The following theorem holds from Theorem 4.1.3 .
When math0220.png are continuous functions on a closed interval I and are the rth derivative functions of f,
the following expression holds for math0221.png.
math0222.png
This theorem guarantees that only the lineal exists in the higher differentiation.
(3) Higher Derivative of Elementary Functions
When m is a natural number, the 2nd order derivative of math0223.pngbecomes as follows.
math0224.png
Then, when math0225.pngis a positive number, it is as follows.
math0226.png
Here, math0227.pngcan be expressed by gamma function math0228.png. Thus
math0229.png
By such an easy calculation, we obtain the following expressions for elementary functions. Where, math0230.png denote the
ceiling function and the floor function respectively.
math0231.png
math0232.png
math0233.png
math0234.png
math0235.png
math0236.png
math0237.png
math0238.png
etc.
(4) Higher Derivative of Inverse Trigonometric Functions
When math0239.png denote the ceiling function and the floor function and n is a natural number,
math0240.png
math0241.png
math0242.png
math0243.png
etc.
(5) Higher Derivative of Inverse Hyperbolic Functions
When math0244.png denote the ceiling function and the floor function and n is a natural number,
math0245.png
math0246.png
math0247.png
etc.
(6) By-Products
math0248.png
math0249.png
Example
math0250.png
math0251.png
10 Termwise Higher Derivative (Trigonometric, Hyperbolic)
In this chapter, for the function which second or more order integral cannot be expressed with the elementary functions
among trigonometric functions and hyperbolic functions, we differentiate the series expansion of these function termwise
and obtain the following expressions. Where, math0252.png denote the ceiling function and the floor function respectively. And
Bernoulli Numbers and Euler Numbers are as follows.
math0253.png
math0254.png
(1) Taylor Series
math0255.png
math0256.png
math0257.png
math0258.png
math0259.png
math0260.png
math0261.png
math0262.png
(2) Fourier Series
math0263.png
math0264.png
math0265.png
math0266.png
(3) Dirichlet Odd Eta (minus) & Even Beta (minus)
Comparing Taylor series and Fourier series, we obtain Dirichlet Odd Eta (minus) and Even Beta (minus) .
math0267.png
math0268.png
math0269.png
math0270.png
math0271.png
math0272.png
(4) Other by-products
math0273.png
math0274.png
math0275.png
math0276.png
math0277.png
math0278.png
math0279.png
math0280.png
11 Termwise Higher Derivative (Inv-Trigonometric, Inv-Hyperbolic)
Among trigonometric functions and hyperbolic functions, there are functions that it is difficult to obtain the general
form of the second or more order derivative. In this chapter, we differentiate the series expansion of these functions
termwise and obtain the following expressions. Where, math0281.png denote the ceiling function and the floor function
respectively.
(1) Taylor Series
math0282.png
math0283.png
math0284.png
math0285.png
math0286.png
math0287.png
math0288.png
math0289.png
math0290.png
math0291.png
math0292.png
math0293.png
(4) By-Products
math0294.png
Example
math0295.png
12 Super Derivative (Non-integer times Derivative)
Here, extending the domain of the index function of a differenciation oprator, we obtain Super Derivative (non-integer
order derivative).
(1) Definitions and Notations
math0296.pngdenotes the non-integer order derivative function of f(x). And we call this Super Derivaive of f(x).
Since there is a super derivative function innumerably, which math0297.pngmeans follows the definition at that time.
We call it Super Differentiation to differentiate a function f with respect to an independent variable continuously.
And it is described as follows.
math0298.png
(2) Fundamental Theorem of Super Differentiation
Let math0299.pngbe an continuous function on the closed interval I and be arbitrary the r-th order derivative
function of f. And let a(r) be a continuous function on the closed interval [0, p].
Then the following expression holds for math0300.png.
math0301.png
Especially, when math0302.png,
math0303.png
(3) Lineal and Collateral
We call the 2nd term of the right sides of these Constant-of-differentiation Function.
And when Constant-of-differentiation Function is 0, we call the left side Lineal Super Differentiation and we callmath0304.png
Lineal Super Derivaive Function .
Oppositely, when Constant-of-differentiation Function is not 0, we call the left side Collateral Super Differentiation and
we call the right side Collateral Super Derivaive Function.
For example,
when math0305.png,
math0306.png
when math0307.png
math0308.png
As seen from this example, the lineal and the collateral exist in the super differentiation unlike the higher differenciation
(4) Riemann-Liouville Differintegral
When the super integral of f(x) is the one with a fixed lower limit, the super derivative of f(x) is obtained by
the following formula. In this formula, the n th order differenciation is subtracted from the n-p th order integration
and the p th order derivative is obtained. Then, this formula is called Riemann-Liouville Differintegral.
math0309.png
Since the left side has lost the operating function, Reimann-Liouville Differintegral of the right side is very important.
(5) Fractional Derivative & Super Derivative
In traditional Fractional Derivative, the super derivative is drawn from Riemann-Liouville Differintegral.
For example, in the case of f(x)= log x, it is as follws.
math0310.png
math0311.png
math0312.png
Long calculation continues.
math0313.png
math0314.png
Because the technique of Fractional Derivative is difficult like this, it is unknown in whether the case of p =1/3 is
calculable in this way.
Also, it is the same as the case of super integral that Riemann-Liouville Differintegral cannot be applied to the non-
integer order derivative such as sin x.
On the other hand, in Super Derivative that I advocates, first of all, we obtain the following higher derivative.
math0315.png
Since differentiation is the reverse operation of integration, reversing the sign of the indexof the integration operator,
we obtain the following immediately.
math0316.png
Substituting p = 1/2 for this,
math0317.png
math0318.png
Thus, we obtain the desired super derivative very easily.
By the way, when p = 2, according to the formulas
math0319.png
it is as follows.
math0320.png
Furthermore, in a similar way, we can obtain the super derivative such as sin x easily.
(6) Super Derivatives of Elementary Functions
In this way, the following super derivatives obtined from " 09 Higher Derivative " .
math0321.png
math0322.png
math0323.png
math0324.png
math0325.png
math0326.png
math0327.png
math0328.png
math0329.png
math0330.png
math0331.png
math0332.png
math0333.png
math0334.png
(7) By-Products
math0335.pngB() is Beta function
math0336.png
math0337.pngB() is Beta function
13 Termwise Super Derivative
The following termwise super derivatives are obtained from " 10 Termwise Higher Derivative (Trigonometric,Hyperbolic)"
and "11 Termwise Higher Derivative (Inv-Trigonometric, Inv-Hyperbolic)" .
Where, math0338.png denote the ceiling function and the floor function respectively. And Bernoulli Numbers and Euler
Numbers are as follows.
math0339.png
math0340.png
math0341.png
math0342.png
math0343.pngcollateral
math0344.pngcollateral
math0345.png
math0346.png
math0347.png
math0348.png
math0349.png
math0350.png
math0351.pngcollateral
math0352.pngcollateral
math0353.png
math0354.pngcollateral
math0355.png
0<x<1collateral
math0356.png
0<x<1collateral
14 Higher and Super Calculus of Logarithmic Integral etc.
Here, the higher integrals and the super calculus of the double logarithm function and the following four functions are
shown.
math0357.png
(1) Higher Integrals of Logarithmic Integral etc.
math0358.png
math0359.png
math0360.png
math0361.png
math0362.png
math0363.png
math0364.png
math0365.png
(2) Super Calculus of Logarithmic Integral etc.
math0366.png
math0367.png
math0368.png
math0369.png
15 Higher and Super Calculus of Elliptic Integral
When math0370.png, Elliptic Integrals of the 1sth3rd kind are expressed as follows.
math0371.png
math0372.png
In this chapter, we expand these to a double series or a triple series and calculate the arc length of an ellipse and
a lemniscate using thiese. Next, we calculate these term by term.
(1) Double (triple) Series Expansion
math0373.png
math0374.png
math0375.png
(2) Termwise Higher Calculus
math0376.png
math0377.png
math0378.png
math0379.png
math0380.png
math0381.png
(3) Termwise Super Calculus
math0382.png
math0383.png
math0384.png
math0385.png
math0386.png
math0387.png
16 Higher Integral of the Product of Two Functions
We obtain the following thorem for the product of two functions.
(1) Theorem 16.1.2
Let math0388.png be the arbitrary rth order primitive function of f(x)and math0389.png be the rth order derivative function of g(x)
for math0390.png. Let math0391.png be the function values of math0392.png on math0393.png.
And let B(n, m) be the beta function. Then, the following formulas hold.
math0394.png
math0395.png
math0396.png
math0397.png(2.1)
Especially, when math0398.png and
when math0399.png,
math0400.png
(2) Higher Integral of the Product of Two Functions
We obtain the following expressions using this theorem.
math0401.png
math0402.png
math0403.png
math0404.png
math0405.png
math0406.png
math0407.png
math0408.png
math0409.png
math0410.png
math0411.png
math0412.png
math0413.png
math0414.png
math0415.png
math0416.png
math0417.png
math0418.png
math0419.png
math0420.png
math0421.png
math0422.png
math0423.png
math0424.png
(3) By-Products
math0425.png
math0426.png
Example
math0427.png
math0428.png
17 Super Integral of the Product of Two Functions
We obtain the following thorems for the product of two functions.
(1) Theorem 17.1.2
Let r, p are positive numbers, math0429.pngbe an arbitrary r th order primitive function of f(x),math0430.png be the r th order
derivative function of g(x), math0431.pngare the beta function and the gamma function respectively. At this
time, if there is a number a such that
math0432.png,
then the following expression holds.
math0433.png
math0434.png
(2) Super Integral of the Product of Two Functions
We obtain the following expressions using this theorem.
math0435.png
math0436.png
math0437.png
math0438.png
math0439.png
math0440.png
math0441.png
math0442.png
math0443.png
math0444.png
math0445.png
math0446.png
math0447.png
math0448.png
math0449.png
math0450.png
math0451.png
math0452.png
math0453.png
math0454.png
(3) By-Products
math0455.png
math0456.png
18 Higher Derivative of the Product of Two Functions
The following Leibniz rule is drawn from the Theorem 16.1.2.
Theorem 18.1.1 (Leibniz)
When functions f(x) and g(x) are ntimes differentiable, the following expression holds.
math0457.png
(2) Higher Derivative of the Product of Two Functions
We obtain the following expressions using this theorem. Where, math0458.pngare real numbers and m is a non-negative
integer. Furthermore, if math0459.png, it shall read as follows.
math0460.png
math0461.png
math0462.png
math0463.png
math0464.png
math0465.png
math0466.png
math0467.png
math0468.png
math0469.png
math0470.png
math0471.png
math0472.png
math0473.png
math0474.png
math0475.png
math0476.png
math0477.png
math0478.png
math0479.png
math0480.png
math0481.png
math0482.png
math0483.png
math0484.png
math0485.png
math0486.png
math0487.png
math0488.png
math0489.png
math0490.png
math0491.png
etc.
(3) By-Products
math0492.png
math0493.png
math0494.png
19 Super Derivative of the Product of Two Functions
The following Leibniz rule for Super Derivative is drawn from the Theorem 16.1.2.
Theorem 19.1.1
Let B(x,y) be the beta function and p be a positive number. And for math0495.png, let math0496.pngbe arbitrary
primitive function of f(x)and math0497.pngbe the rth order derivative function of g(x). Then the following expressions hold
math0498.png
math0499.png
Especially, when math0500.png
math0501.png( Leibniz )
(2) Super Derivative of the Product of Two Functions
We obtain the following expressions using this theorem. Where, math0502.pngare real numbers and m is a non-negative
integer. Furthermore, if math0503.png, it shall read as follows.
math0504.png
math0505.png
math0506.png
math0507.png
math0508.png
math0509.png
math0510.png
math0511.png
math0512.png
math0513.png
math0514.png
math0515.png
math0516.png
math0517.png
math0518.png
math0519.png
math0520.png
math0521.png
math0522.png
math0523.png
math0524.png
math0525.png
math0526.png
math0527.png
math0528.png
math0529.png
math0530.png
math0531.png
math0532.png
etc.
(3) By-Products
math0533.png
math0534.png
20 Higher Calculus of the product of many functions
20.1 Higher Derivative of the product of many functions
The following theorems are drawn from the Theorem 18.1.1.
Theorem 20.1.1
When math0535.pngdenotes the r th order derivative function of math0536.png,
math0537.png
Theorem 20.1.2
When math0538.pngdenotes the r th order derivativefunction of f(x) and math0539.pngis a natural number,
math0540.png
Example
math0541.png
math0542.png
math0543.png
Higher Derivatives of math0544.png
math0545.png
math0546.png
math0547.png
math0548.png
20.2 Higher Integral of the product of many functions
The following theorems are drawn from the Theorem 16.1.2 and Theorem 20.1.1.
Theorem 20.2.1
Let math0549.pngbe the r th order derivative function of math0550.png, math0551.png be the arbitrary r th order primitive
function of math0552.png, m, n are natural numbers and B(n,m)be the beta function. If there is a number a such that
math0553.pngor math0554.png for at least one k>1, then the
following expression holds.
math0555.png
math0556.png
math0557.png
math0558.png
Theorem 20.2.2
Let math0559.pngbe the r th order derivative function of f(x), math0560.pngbe the arbitrary r th order primitive function of f(x),
m, n are natural numbers and B(n,m)be the beta function. At this time , if there is a number a such that
math0561.png
then the following expression holds for math0562.png.
math0563.png
math0564.png
math0565.png
Example
math0566.png
math0567.png
math0568.png
math0569.png
math0570.png
Higher Integrls of math0571.png
math0572.png
math0573.png
math0574.png
math0575.png
Where, math0576.png are the solutions of the following transcendental equations.
math0577.png
math0578.png
math0579.png
math0580.png
math0581.png
math0582.png
21 Super Calculus of the product of many functions
21.1 Super Integrals of the product of many functions
The following theorems are drawn from the Theorem 20.2.1.
Theorem 21.1.1
Let p, r are positive numbers, m be a natural number, math0583.pngbe the rth order derivative function of
math0584.png, math0585.pngbe arbitrary r th order primitive function of math0586.png and B(p,m) be
the beta function. At this time, if there is a number a such that
math0587.png or math0588.png for at least one k>1,
the following expression holds
math0589.png
math0590.png
math0591.png
Theorem 21.1.2
Let p, r are positive numbers, m be a natural number, math0592.pngbe the rth order derivative function of f(x),
math0593.png be arbitrary math0594.pngth order primitive function of f(x) and B(p,m) be the beta function. At this time,
if there is a number a such that
math0595.png or math0596.png
the following expression holds for math0597.png.
math0598.png
math0599.png
math0600.png
Example
math0601.png
math0602.png
math0603.png
math0604.png
math0605.png
Super Integrls of math0606.png
math0607.png
math0608.png
Where, math0609.pngare zeros of lineal super primitives of math0610.pngor math0611.png.
21.2 Super Derivatives of the product of many functions
The following theorems are drawn from the Theorem 21.1.1.
Theorem 21.2.1
Let p, r are positive numbers, m be a natural number, math0612.pngbe the rth order derivative function of
math0613.png, math0614.pngbe arbitrary r th order primitive function of math0615.png and B(p,m) be
the beta function. At this time, if there is a number a such that
math0616.png or math0617.png for at least one k>1,
the following expression holds
math0618.png
math0619.png
math0620.png
Theorem 21.2.2
Let p, r are positive numbers, m be a natural number, math0621.pngbe the rth order derivative function of f(x),
math0622.png be arbitrary math0623.pngth order primitive function of f(x) and B(p,m) be the beta function. At this time,
if there is a number math0624.png such that
math0625.png or math0626.png
the following expression holds for math0627.png.
math0628.png
math0629.png
math0630.png
Super Derivatives of math0631.png
math0632.png
math0633.png
22 Higher Derivative of Composition
22.1 Formulas of Higher Derivative of Composition
About the formula of the higher derivative of composition, the following formula was shown by Faà di Bruno about 150
years ago.
Formula 22.1.1 ( Faà di Bruno )
Let math0634.png are non-negative integers. Let math0635.pngare derivative functions and math0636.png are
the 2nd kind of Bell polynomials such that
math0637.png
math0638.png
math0639.png
Then, the higher derivative function with respect to x of the composition g{f(x)} is expressed as follows.
math0640.png
Next, the algorithm that generates math0641.pngwithout an omission is shown. And the derivatives up to
the 8 th order of z= g{f(x)} are calculated using this.
Next, trying the higher differentiation of some compositions, we obtain the following formulas.
math0642.png
math0643.png
math0644.png
math0645.png
Even if the algorithm above is used, it is not so easy to obtain math0646.png.
However, when the core function f(x)is the 1st degree, it becomes remarkably easy.
Formula 22.1.3
When math0647.png , if f(x) is the 1st degree, the higher derivative function
with respect to x of the composition g{f(x)} is expressed as follows.
math0648.png
And, this can be easily enhanced to the super-differentiation in this case. That is, the following expression holds
for p>0 . This is the grounds for which we have used "Linear form" since "12 Super Deribative" as a fait accompli .
math0649.png
23 Higher Integral of Composition
Formula 23.1.1
Let math0650.pngbe the lineal higher primitive function of g(f) and math0651.pngare the functions of f such that
math0652.png
Let math0653.png are the polynomials such taht
math0654.png
math0655.png
math0656.png
And let math0657.png are the zeros of the lineal higher primitive functions of g{f(x)} , gh respectively.
Then, the lineal higher integral with respect to x of the composition g{f(x)} is expressed as follows.
math0658.png
math0659.png(n-fold nest)
math0660.png
math0661.png
math0662.png
math0663.png
If it writes down up to the 3rd order without using math0664.png, it is s follows.
math0665.png
math0666.png
math0667.png
math0668.png
math0669.png
math0670.png
math0671.png
math0672.png
math0673.png
math0674.png
math0675.png
math0676.png
Next, trying the higher integration of some compositions, we obtain the following formulas
math0677.png
math0678.png
math0679.png
math0680.png
math0681.png
math0682.png
math0683.png
math0684.png
Formula 23.1.1 is so complicated and is not applicable to any composition. The 1st, the inverse function of the core
function must be known. The 2nd, the higher primitive function of the enclosing function g(f) must have the property
such as math0685.png. Considering these, there are not many compositions that Formula 23.1.1 is applicable.
However, when the core function f(x) is the 1st degree, it becomes remarkably easy.
Formula 23.1.2
When f(x)= cx+d,
math0686.png
And, this can be easily enhanced to the super integral in this case. That is, the following expression holds for p>0.
This is the grounds for which we have used "Linear form" since "07 Super Integral" as a fait accompli.
math0687.png
24 Sugioka's Theorem on the Series of Higher (Repeated) Integrals
Mikio Sugioka showed the following in his work " math0688.png ".
1. The series of the higher integral of a function f(x) results in one integral of math0689.png, sin x, cos x, sinh x, cosh x,
2. The series of the higher integral of a function f(x) is expanded into the series of the higher integrals of math0690.png, etc.
These are introduced and proved in this chapter. In this summary, we assume math0691.png. Also,
we describe the n th order differential coefficient at a with math0692.png
(1) Sum of the Series of Higher Integrals
math0693.png
math0694.png
math0695.png
math0696.png
math0697.png
math0698.png
Example
math0699.png
math0700.png
math0701.png
math0702.png
math0703.png
(2) Integrals Series Expansion
math0704.png
math0705.png
math0706.png
math0707.png
math0708.png
math0709.png
Example
math0710.png
math0711.png
math0712.png
math0713.png
math0714.png
math0715.png
math0716.png
math0717.png
(3) Series of Higher Integrals with coefficients
math0718.png
math0719.png
Example
math0720.png
math0721.png
math0722.png
math0723.png
math0724.png
2011.04.16
2012.10.28 Renewal
K. Kono
Alien's Mathematics