Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Power Means Calculus Product Calculus,
Harmonic Mean Calculus, and Quadratic Mean Calculus H. Vic Dannon [email protected] March, 2008
Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Abstract 1 aarr++... a rr Each Power Mean of order r ≠ 0, 12 n ( n ) is associated with a Power Mean Derivative of order r , D()r . We describe the Arithmetic Mean Calculus obtained if r = 1, Geometric Mean Calculus obtained if r → 0 , Harmonic Mean Calculus obtained if r =−1, Quadratic Mean Calculus obtained if r = 2
Keywords Calculus, Power Mean, Derivative, Integral, Product
Calculus. Gamma Function,
Mathematics Subject Classification 26A06, 26B12, 33B15,
26A42, 26A24, 46G05,
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Contents
Introduction..…………………………………………………………………………9
1. Arithmetic Mean Calculus...... 11
1.1 The Arithmetic Mean of f()x over [,]ab…………………………………...... 11
1.2 Mean Value Theorem for the Arithmetic Mean…………………………...... 12 1.3 The Arithmetic Mean of f()x over [,xx+ dx ]…………………...... 12
1.4 Arithmetic Mean Derivative……………………………………………………13 1.5 The Arithmetic Mean Derivative is the Fermat-Newton-Leibnitz Derivative……….…………………………………………………………………13 1.6 The Arithmetic Mean Derivative is an Additive Operator……………...... 14 1.7 The Arithmetic Mean Derivative is not a Multiplicative Operator………14
2. The Product Integral…………………………………………………………..15
2.1 Growth problems…………………………………………………………………15
2.2 The Product Integral of ert() over [,]ab……………………………..……….15
2.3 The Product Integral of f()x over [,]ab………………………….…………..16
2.4 Intermediate Value Theorem for the Product Integral…………………….17 2.5 The Product Integral is a Multiplicative Operator…………………………18
3. Geometric Mean and Geometric Mean Derivative...... 19
3.1 The Power Mean with r → 0 is the Geometric Mean……………..……...19 3.2 The Geometric Mean of f()x over [,]ab……………………………..……...20
3.3 Mean Value Theorem for the Geometric Mean……………………………..20 3.4 The Geometric Mean of f()x over [,xx+ dx ]……………………………...21
3.5 Geometric Mean Derivative……………………………………………………21
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3.6 DGx(0)()= eDGx log ( ) …………………………………………………………..22
DG() x 3.7 DGx(0)()= eGx ( ) ……………………………………………………………..22
3.8 The Geometric Mean Derivative is non-additive operator………………..22 3.9 The Geometric Mean Derivative is multiplicative operator………………23 3.10 Geometric Mean Derivative Rules……………………………………..…….23
4. Geometric Mean Calculus……………………………………………….…...24
4.1 Fundamental Theorem of the Product Calculus………………………..…...24 4.2 Table of Geometric Mean Derivative, and Product Integrals……………..24
5. Product Differential Equations……………………………………..………26
5.1 Product Differential Equations……………………………………….………..26
5.2 Product Calculus Solution of dy = Pxy()……………………………...…..27 dx
5.3 dy =+Pxy() Qx () may not be solved by Product Calculus.…………....27 dx 5.4 yPxyQxy''=+ ( ) ' ( ) may not be solved by Product Calculus...... 29
6. Product Calculus of sin x …………………………….…………………….....30 x
6.1 Euler’s Product Representation for sin z …..…………………………………30 z 6.2 Conversion to Trigonometric Series…………………………………..……….30
6.3 Geometric Mean Derivative of sin x ..……………………………...…………..31 x
6.4 Second Geometric Mean Derivative of sin x ..……………………….………..32 x
6.5 Product Integration of sin x ..……………………………………………………32 x
6.6 Euler’s 2nd Product Representation for sin z …..……………………………..34 z
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6.7 Geometric Mean Derivative of Euler’s 2nd product for sin x ...... 35 x
7. Product Calculus of sin x ………………………..……………………………36
7.1 Euler’s Product Representation for sin z …..……………………...………...36 7.2 Geometric Mean Derivative of sin x ………………………………...………..39 7.3 The Wallis Product for π ……………………………………………………….39
8. Product Calculus of cosx …………………………………………...………..41 8.1 Euler’s Product Representation for cosz ………………………………..…..41 8.2 Geometric Mean Derivative of cosx ……………………………..…………..41
9. Product Calculus of tanx …………………………………………..………..43 9.1 Product Representation for tanz ……………...……………………..….……43 9.2 Geometric Mean Derivative of tanx ……………………………..…….…….43
10. Product Calculus of sinhx ………………………..……………..…………45
10.1 Product Representation of sinh z ………………………………...………….45 10.2 Geometric Mean Derivative of sinh x ……………………………..………..45
11. Product Calculus of cosh x …………………………………...……….……46
11.1 Product Representation of cosh z ……………………………………………46 11.2 Geometric Mean Derivative of cosh x ……………………………………....46
12. Product Calculus of tanhx ………………………………….....……..……47
12.1 Product Representation for tanhx ……………………………..…………..47 12.2 Geometric Mean Derivative of tanhx …………………………..………….47
13. Product Calculus of ex ………………………..………………………..……49
13.1 Product representation of ex ………………………………………………….49
13.2 Geometric Mean Derivative of ex ……………………………………………49
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x 13.3 Geometric Mean Derivative of ee ……………………………………..…...50
k 13.4 Geometric Mean Derivative of ex …………………………………………..50
14. Geometric Mean Derivative by Exponentiation…………………..…52
14.1 Dxe(0)sin = cotx ………………………………………………………………52
14.2 Dx(0) x = ex………………………………………………………….………...53
15. Product Calculus of Γ()x ……………………………………………………55
15.1 Euler’s Product Representation for Γ()x …………………………………...56
15.2 Geometric Mean Derivative of Γ()x …………………………………………58
15.3 Γ=(1) 1 ………………………………………………………………………….60
15.4 Γ+(1)()zzz =Γ ………………………………………………………………60
15.5 Product Reflection Formula for Γ()z ……………………………………….61
15.6 ΓΓ−=()(1zz ) π …………………………..……………………………...61 sin πz
2 15.7 Γ=()1 π ……………………………………………………………………62 ()2
2 15.8 ⎡⎤Γ=⋅12 2244668810101212 ....……………………….……..……62 ⎣⎦⎢⎥( 2) 13355779 9 111113
16. Products of Γ()z ………………………………………………………………64
Γ+(1z1 ) 16.1 , where zww112=+...... 65 Γ+(1ww12 ) Γ+ (1 )
Γ(1) 16.2 = sinhx ...... 65 Γ+(1ix ) Γ− (1 ix )
Γ+(1zz12 ) Γ+ (1 ) 16.3 , where zz12+= www 1 + 2 + 3……………….66 Γ+(1www123 ) Γ+ (1 ) Γ+ (1 )
Γ+(1zz ) Γ+ (1 )... Γ+ (1 z ) 16.4 12 k ...... 66 Γ+(1ww12 ) Γ+ (1 ).... Γ+ (1 wl )
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17. Product Calculus of Jxν()………………………………………………..…67
17.1 Product Formula for Jzν ()……………………………………………………67
17.2 Geometric Mean Derivative of Jzν ()……………………………………….67
18. Product Calculus of Trigonometric Series………………….…………69
18.1 Product Integral of a Trigonometric Series…………………………………69
19. Infinite Functional Products………………………………………………70
19.1 Geometric Mean Derivative of an Euler Infinite Product……………….70
20. Path Product Integral………………………………………………………..71
20.1 Path Product Integral in the Plane…………………………………………..71 20.2 Green’s Theorem for the Path Product Integral………..………………….71
20.3 Path Product Integral in E 3 ………………………………………………….72 20.4 Stokes Theorem for the Path Product Integral…………………………….72
21. Iterative Product Integral…………………………………………………..73
21.1 Iterative Product Integral of f(,)xt ………………………………………..73
21.2 Iterative Product Integral of erxtdx(,) ...... 73
22. Harmonic Mean Integral…………………………………………………….74
22.1 Harmonic Mean Integral………………………………………………………74
23. Harmonic Mean and Harmonic Mean Derivative……………………75
23.1 The Harmonic Mean of f ()x over [,]ab...... 75
23.2 Mean Value Theorem for the Harmonic Mean…………………………….76 23.3 The Harmonic Mean of f ()x over [,xx+ dx ]...... 76
23.4 Harmonic Mean Derivative……………………………………………………77
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23.5 DHx(1)− ()= 1 ….………………………………………………………..77 DHx () x
24. Harmonic Mean Calculus……………………………………………………78
24.1 The Fundamental Theorem of the Harmonic Mean Calculus…………...78 24.2 Table of Harmonic Mean Derivatives and Integrals………………………78
25. Quadratic Mean Integral.…………………………………………………...80
25.1 Quadratic Mean Integral……………………………………………………...80 25.2 Cauchy-Schwartz Inequality for Quadratic Mean Integrals……….……81 25.3 Holder Inequality for Quadratic Mean Integrals …………………………81
26. Quadratic Mean and Quadratic Mean Derivative…………………...83
26.1 Quadratic Mean of f()x over [,]ab...... 83
26.2 Mean Value Theorem for the Quadratic Mean…………………………….84 26.3 The Quadratic Mean of f()x over [,xx+ dx ]...... 84
26.4 Quadratic Mean Derivative…………………………………………………..84
(2) 1/2 26.5 DQx()= ( DQxx ()) ……………………………………………………..85
27. Quadratic Mean Calculus…………………………………………………..86
27.1 The Fundamental Theorem of the Quadratic Mean Calculus…………..86 27.2 Table of Quadratic Mean Derivatives and Integrals……………………..86
References……………………………………………………………………………88
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Introduction
We describe a generalized calculus that was suggested by Michael Spivey’s [Spiv] observation of the relation between the Geometric Mean of a function over an interval, and its product integral. We will see that each Power Mean of order r ≠ 0,
1 aarr++... a rr 12 n ( n ) is associated with a Power Mean Derivative of order r ,
D()r . The Fermat/Newton/Leibnitz Derivative
d DD(1) =≡ dx is associated with the Arithmetic Mean
aa++... a 12 n , n which is Power Mean of order r = 1. The Geometric Mean Derivative
D(0) is associated with the Geometric Mean
1/n (aa12... an ) which is Power Mean of order r → 0 [Kaza].
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Product Integration is an operation inverse to the Geometric Mean Derivative. Both are multiplicative operations, that apply naturally to products, and in particular to Γ()z , the analytic extension of the factorial function The Harmonic Mean Derivative
D(1)− is associated with the Harmonic Mean
n 11 1 ++... aa12 an which is Power Mean of order r =−1. The Quadratic Mean Derivative
D(2) is associated with the Power Mean of order r = 2 ,
1 aa22++... a 22 12 n . ( n )
The inverse operation, the Quadratic Mean Integration transforms a function to its L2 norm squared. We proceed with the definition of the Arithmetic Mean Derivative.
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Arithmetic Mean Calculus
1.1 The Arithmetic Mean of f()x over [,]ab
Given a function f()x that is Riemann integrable over the interval
[,]ab, partition the interval into n sub-intervals, of equal length
ba− Δ=x , n choose in each subinterval a point
ci , and consider the Arithmetic Mean of f()x ,
f ()cfcfc++ () ...() 1 12 n =++Δ()f()cfcfcx () ...() nba− 12 n
As n →∞, the sequence of the Arithmetic Means converges to
xb= Fb()− Fa () 1 fxdx() = , ba−−∫ ba xa= where tx= Fx()= ∫ ftdt () . t =0 Therefore, the Arithmetic Mean of f()x over [,]ab is defined by
xb= 1 fxdx() ba− ∫ xa=
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1.2 Mean Value Theorem for the Arithmetic Mean xb= There is a point acb<<, so that 1 f()xdx= fc (). ba− ∫ xa= tx= Proof: Since Fx()= ∫ ftdt () is continuous on [,]ab, and t =0 differentiable in (,)ab , by Lagrange Intermediate Value Theorem there is a point acb<<, so that Fb()− Fa () = f ()c . ba− That is, xb= 1 f ()xdx= fc (). ba− ∫ xa=
1.3 The Arithmetic Mean of f()x over [,xx+ dx ]
The Arithmetic Mean of f ()x over [,xx+ dx ] is the Inverse
operation to Integration Proof: By 1.2, there is xcx<<+Δ x, so that tx=+Δ x 1 f ()tdt= fc () Δx ∫ tx= Letting Δ→x 0, the Arithmetic Mean of f()x at x , equals f()x .
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tx=+Δ x lim1 f (tdt )= f ( x ). Δ→x 0 Δx ∫ tx= Thus, the operation of finding the Arithmetic Mean of f ()x x , is inverse to integration.
This leads to the definition of the Arithmetic Mean Derivative.
tx= 1.4 Arithmetic Mean Derivative of Fx()= ∫ ftdt () at x t =0 The Arithmetic Mean Derivative of
tx= Fx()= ∫ ftdt () t =0 at x is defined as the Arithmetic Mean of f()x over [,xx+ dx ]
tx=+Δ x DFx(1) ( )≡ lim1 ftdt ( ) Δ→x 0 Δx ∫ tx=
1.5 The Arithmetic Mean Derivative is the Fermat-
Newton-Leibnitz derivative
DFx(1) ()= dF() x dx
txdx=+ Proof: DFx(1) ( )= Standard Part of 1 ftdt ( ) dx ∫ tx= = Standard Part of Fx()()+− dx Fx dx
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==dF() x DF() x . dx
1.6 The Arithmetic Mean Derivative is an Additive
Operator
DFx()12()+= Fx () DFx 1 () + DFx 2 ()
Thus, the Arithmetic Mean Derivative applies effectively to infinite series.
1.7 The Arithmetic Mean Derivative is not a multiplicative operator
DFxFx( 12() ()) =+( DFxFx 1 ()) 2 () Fx 1 ()( DFx 2 ())
Thus, Arithmetic Mean Derivative does not apply easily to infinite products.
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The Product Integral
2.1 Growth Problems
The Arithmetic Mean Derivative is unsuitable when we are interested in the quotient
Present Value
Invested Value Similarly, attenuation or amplification is measured by
Out-Put Signal . In-Put Signal The need for a multiplicative derivative operator motivated the creation of the product integration.
2.2 The Product Integral of ert() over the interval [,]ab
An amount A compounded continuously at rate rt() over time dt becomes Aertdt() . Over n equal sub-intervals of the time interval [,]ab,
Δ=t ba− , n we obtain the sequence of finite products
rt()ΔΔ t rt () t rt () Δ t [() rt +++Δ rt ()...()] rt t Ae12 e... enn= Ae 12 .
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As n →∞, the sequence converges to
tb= ∫ rtdt() Aeta= The amplification factor
tb= ∫ rtdt() eta= is called the product integral of ert() over the interval [,]ab and is denoted tb= ∏ertdt() . ta=
Thus, the Product Integral of ert() over the interval [,]ab is
tb= tb= ∫ rtdt() ∏eertdt() ≡ ta= ta=
2.3 The Product Integral of f()x over the interval [,]ab
Given a Riemann integrable, positive f()x on [,]ab, partition the interval into n sub-intervals, of equal length
ba− Δ=x , n choose in each subinterval a point
ci , and consider the finite products,
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ΔΔxx Δ x ΔΔxx Δ xln[fc (12 ) fc ( ) ... fc (n ) ] f ()cfcfce12 () ...()n =
[lnf (cfcfcx )+++Δ ln ( ) ... ln ( )] = e 12 n . As n →∞, the sequence of products converges to
xb= ∫ ()lnfx ( ) dx exa= > 0 . We call this limit the product integral of f ()x over the interval [,]ab, and denote it by xb= ∏ f()x dx . xa= Thus, the Product Integral of f()x over the interval [,]ab is
xb= xb= ∫ ()lnf (xdx ) ∏ fx()dx ≡ exa= xa=
2.4 Intermediate Value Theorem for the Product Integral
xb= There is a point acb<<, so that ∏ fx()dx= fc ()() b− a xa=
tx= Proof: Since ϕ()xftdt= ∫ () ln() is continuous on [,]ab, and t =0 differentiable in (,)ab , by Lagrange Intermediate Value Theorem there is a point
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acb<<, so that
xb= ∫ ()lnf (xdx )=−=ϕϕ ( b ) ( a ) () ln fc ( ) ( b − a ). xa= Hence,
xb= ∫ ()lnfxdx ( ) eexa= ==()lnfc ( ) ( b− a ) f()c ()ba− .
2.5 The Product Integral is a multiplicative operator xb===⎛⎞⎛⎞ xc xb ⎜⎜⎟⎟ If acb<<, fx()dx= ⎜⎜ fx () dx⎟⎟ fx () dx ∏∏∏⎜⎜⎟⎟ xa===⎝⎠⎝⎠⎜⎜ xa⎟⎟ xc
Proof: If acb<<,
xc== xb xb===()lnfx ( ) dx+ () ln fx ( ) dx ⎛⎞⎛⎞ xc xb ∫∫ ⎜⎜⎟⎟ fx()dx== exa== xc ⎜⎜ fx () dx⎟⎟ fx () dx . ∏∏∏⎜⎜⎟⎟ xa===⎝⎠⎝⎠⎜⎜ xa⎟⎟ xc
The inverse operation to product integration is the Geometric Mean Derivative.
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Geometric Mean and Geometric
Mean Derivative
3.1 The Power Mean with r → 0 is the Geometric Mean 1 1 aarr++... a rr 12 n n → ()aa12... a 3 ( n ) r→0 Proof: Let r → 0 in 1 rr r 1 ⎡ rr r ⎤ aa++... ar log()aa12++ ... an − log n 12 n = e r ⎣⎢ ⎦⎥ . ( n )
Then, the exponent 1 ⎡⎤log(aarr++ ... a r ) − log n is of the form 0 , r ⎣⎦⎢⎥12 n 0 and by L’Hospital, its limit is
rr r Daaarn{}log(12++ ... ) − log n lim r →0 Drr
1 rr r =++limrr r(aaaa1122 ln ln ... aann ln ) r →0 aa12++... an
=+++1 lnaa ln ... ln a n ( 12 n )
1 n = ln(aa12 ... an ) . Therefore, 1 1 1 aarr++... a rr ln(aa ... a )n 12 n ⎯⎯⎯→eaaa12 n = ... n . ( n ) r →0 ()12 n
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3.2 Geometric Mean of f()x over [,]ab
Given an integrable function f()x that is positive over [,]ab, partition the interval, into n sub-intervals, of equal length
ba− Δ=x , n choose in each subinterval a point
ci , and consider the Geometric Mean of f()x
1 1/n ()lnf (cfcfcx )++ ln ( ) ...ln ( ) Δ ba− 12 n ()fc(12 ) fc ( )... fc (n ) = e . As n →∞, the sequence of Geometric Means converges to
xb= 1 ()lnfx ( ) dx ba− ∫ Gb() e xa= = , Ga() where tx= ∫ ()lnf (tdt ) Gx()= eta= Therefore,
xb= 1 ()lnfx ( ) dx ba− ∫ e xa= is defined as the Geometric Mean of f()x over [,]ab.
3.3 Mean Value Theorem for the Geometric Mean
xb= 1 ()lnfx ( ) dx ba− ∫ There is a point acb<<, so that efcxa= = ()
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Proof: By 2.3, and 2.4.
3.4 The Geometric Mean of f()x over [,xx+ dx ]
The Geometric Mean of f()x over [,xx+ dx ], is the Inverse
Operation to Product Integration Proof: By 3.3, there is xcx<<+Δ x, so that
tx=+Δ x 1 lnftdt ( ) Δx ∫ efctx= = ().
Letting Δ→x 0, the Geometric Mean of f()x at x equals f ()x .
tx=+Δ x 1 lnftdt ( ) Δx ∫ limefxtx= = ( ) Δ→x 0 Thus, the operation of finding the Geometric Mean of f ()x over
[,xx+ dx ], is inverse to product integration over [,xx+ dx ].
This leads to the definition of the Geometric Mean Derivative
3.5 Geometric Mean Derivative
The Geometric Mean Derivative of
tx= ∫ ()lnf (tdt ) Gx()= eta=
at x is defined as the Geometric Mean of f ()x over [,xx+ dx ]
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tx=+Δ x 1 lnft ( ) dt Δx ∫ () DGx(0) ()≡ lim e tx= Δ→x 0
3.6 DGx(0)()= eDGx log ( )
tx=+Δ x 1 lnf (tdt ) Δx ∫ () Proof: DGx(0) ()= lim e tx= Δ→x 0 tx=+Δ x lim1 () lnf (tdt ) Δ→x 0 Δx ∫ = e tx= txdx=+ 1 lnf (tdt ) dx ∫ () = Standard part of e tx=
1 = Standard Part of Gx()+ dx dx ()Gx()
1 ⎡ logGx (+− dx ) log Gx ( )⎤ = Standard Part of e dx ⎣ ⎦
= eDGxlog ( ) .
DG() x 3.7 DGx(0)()= eGx ( )
3.8 The Geometric Mean Derivative is non- additive operator
DG()12() x+ G () x
(0) Gx12()+ Gx () Proof: DG()()12+= Gx e .
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3.9 The Geometric Mean Derivative is a multiplicative operator
DGxGxlog[ ( ) ( )] (0) ⎡⎤12 Proof: DGxGx⎣⎦12() ()= e
DG() x DG () x 12+ ()Gx() Gx () = e 12
DG12() x DG () x = eeGx12() Gx ()
(0) (0) = (DGx12())( DGx ()).
3.10 Geometric Mean Derivative Rules
2 2 ()DGxDe(0)()≡= (0) DGxln ( ) e D ln Gx ( )
n n (DGxe(0) ) ()= DGxln ( )
Dfx(0)( ()gx ( )) = fxe () Dgx ( ) gxD ( ) ln fx ( )
df dg dg dx Dfgx(0)(())= efgx ( ( ))
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Geometric Mean Calculus
4.1 The Fundamental Theorem of the Product Calculus
tx= Dftfx(0)∏ ()dt = ( ) ta=
tx= tx= ∫ ()lnf (tdt ) (0)dt (0) ta= Proof: DftDexx∏ () = ta=
tx= 1 ∫ ()lnft ( ) dt Deta= tx= x ∫ ()lnft ( ) dt = eeta=
tx= ∫ ()lnft ( ) dt tx= ta= eDftdtx ∫ ()ln ( ) ta= tx= ∫ ()lnft ( ) dt = e eta=
tx=
Dftdtx ∫ ()ln ( ) = e ta=
= f()x .
4.2 Table of Geometric Mean Derivatives and integrals
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We list some geometric Mean Derivatives, and Product Integrals. Some of these are given in [Spiv].
fx()DIfx(0) fx () (0) ()= ∏ fx () x 11 1 21 2x aa1 x ee1 x xe1/xxx e (ln− 1) = ex−xx xe2222/xxx e 2 (ln− 1) = ex− xx xea ax/(ln1) e ax x− = ex−ax ax xe−1 −−−1/xxx e (ln 1) = exxx lnx e1/xx log e∫ ln() lnxdx ee22xx e2 eeax a eax2/2 xx22 2 − xexex xx(ln− 1/2)/2 = e 42x eexnxnn−1 exnn+1/(+ 1) ln(sinxdx ) sinxecotx e∫ ln(cosxdx ) cosxe−tanx e∫ ln(tanxdx ) tanxe2/sin2x e∫ eeeexx e e x eesinxx cos e− cos x cosxx− sin sin x ee e
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Product Differential Equations
5.1 Product Differential Equations
A Product Differential Equation involves powers of the Geometric Mean Derivative operator
D(0) , and no sums, only products. An ordinary differential equations involves sums of powers of the Arithmetic Mean Derivative operator D . Such equation is not suitable to the application of D(0) , and does not convert easily into a product differential equation. [Doll] attempts to write the solutions to ordinary differential equations in terms of product integral, but being unaware of the Geometric Mean Derivative, it fails to produce one product differential equation. [Doll] demonstrates that products integrals are not natural solutions for ordinary differential equations. The attempt made in [Doll] to interpret Summation Calculus in terms of the Product Integral alone, being oblivious to the product Calculus derivative, does not lead to better understanding of differential equations, or to any new results.
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Only the basic equation
dy = Pxy() dx may be converted to a product differential equation, and be solved as such.
dy 5.2 Product Calculus Solution of = Pxy(). dx Dividing both sides by yx(),
y ' = Px(). y
y ' eey = Px() Dy(0)= ePx ( )
tx= tx= ∫ Ptdt() yx()==∏eePtdt() t =0 . t =0
dy 5.3 =+Pxy() Qx () may not be solved by Product Calculus dx Proof: We don’t know of a Product Calculus method to solve the equation
dy =+Pxy() Qx (). dx
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tx= ∫ Ptdt() In Arithmetic Mean Calculus, we multiply both sides by et =0 . Then,
tx=== tx tx ∫∫∫P() t dt P () t dt P () t dt ye'()()ttt===000+= yPxe Qxe
tx== tx d ∫∫Ptdt() Ptdt () ()()yett==00= Q x e dx
tx== tu ∫∫Ptdt()ux= Ptdt () yett==00= ∫ Q() u e u=0
tu= ux= ∫ Ptdt() ∫ Que()t =0 u=0 y = tx= ∫ Ptdt() et =0 Writing this as
ux= tx= ∫ Qu()∏ePtdt() y = u=0 t =0 tx= ∏ePtdt() t =0 demonstrates why the equation cannot be converted into a product differential equation, and cannot be solved as such: In product Calculus we need to have pure products. No summations.
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5.4 yPxyQxy''=+ ( ) ' ( ) may not be solved by Product Calculus
Proof: We may write yPxyQxy''=+ ( ) ' ( ) as a first order system yz' = zPxzQxy'()()=+
Thus, in matrix form,
d ⎛⎞yy ⎛01 ⎞⎛⎞ ⎜⎜⎟⎟⎟ ⎜ ⎜⎜⎟⎟⎟= ⎜. dx ⎝⎠⎜⎜zQxPxz⎟⎟⎟ ⎝() () ⎠⎝⎠ ⎜
But we need the methods of summation Calculus, to obtain two independent solutions yx1(), and yx2() that span the solution space for the equation.
29 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
6 sinx Product Calculus of x
sin z 6.1 Euler’s Product Representation for z sin z For any complex number z , = coszzz cos cos ... z 248
Proof: sinz = 2 coszz sin 22
= 2 coszzz 2 cos sin 244
= 2coszzzz 2cos 2cos sin 2488
= 2 coszzz 2 cos 2 cos ...2 cos zz sin 248 22nn
= z 2n sinzzzzz cos cos cos ...cos . ( z 22nn) 248 Therefore, for any complex number z ≠ 0,
z ⎛⎞sin z n ⎜ ⎟ 2 = coszzz cos cos ...cos z ⎝⎠⎜ z ⎟ sin z 248 2n 2n Letting n →∞, sinz = coszzz cos cos ... z 248 This holds also for z → 0 . Hence, it holds for any complex number z .
6.2 Conversion to Trigonometric Series
30 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Products of Cosines can be converted into summations, and the infinite product may be converted into a Trigonometric Series. For instance, cosαβγ cos cos=++−11 cos( αβ ) cos( αβ ) cos γ ( 22)
=+11cos(αβ )cos γ +− cos( αβ )cos γ 22
=++++−11cos(αβγ ) cos( αβγ ) 44
+−++−−11cos(αβγ ) cos( αβγ ) 44
sinx 6.3 Geometric Mean Derivative of x
cosx 1 −=−11tanxx − tan − 1 tan x − ... sin xx 2222222233 Proof: Geometric Mean Differentiating both sides of 6.1,
sin x DDDD(0)= ( (0)cosxxx)( (0) cos)( (0) cos) ... x 248
⎛⎞sin x ⎛⎞⎛⎞xx ⎜D ⎟ ⎛⎞x ⎜⎜DDcos⎟⎟ cos ⎜ x ⎟ D cos ⎜⎜23⎟⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜⎜22⎟⎟ ⎜ sin x ⎟ ⎜ ⎟ ⎜⎜⎟⎟ ⎜ ⎟ ⎜ x ⎟ ⎜⎜xx⎟⎟ ⎝⎠⎜ x ⎜ cos ⎟⎟⎟⎟ ⎜⎜cos⎟⎟ cos eeee= ⎝⎠⎝⎠⎝⎠2223 2...
11xx cosxx− 1− 1 tan −−tan tan eeeesinxx= 2 2 22233 2 2 2 That is,
cosx 1 −=−11tanxx − tan − 1 tan x − ... sin xx 2222222233
31 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
sinx 6.4 Second Geometric Mean Derivative of x
11 1 1 1 −+=−− − .... sin22xx xx x 222 cos 2 42 cos 2 62 cos 2 2223
Proof: Second Geometric Mean Differentiation of 6.1 gives
⎛⎞ ⎜ cosx 1 ⎟ ⎜ − ⎟ 1 x − 1 tan x ⎜ ⎟ ()− tan ( 22) De(0)⎝⎠sin xx =× De (0)22 De (0) 2 2 ×....
⎛⎞⎛⎞ ⎛⎞⎜⎜11⎟⎟ ⎜ 1 ⎟ ⎜⎜⎟⎟ 11 ⎜− ⎟ ⎜⎜−−⎟⎟ −+ ⎜ 22x ⎟ ⎜⎜242 cosxx⎟⎟ 2 62 cos ()22 ⎝⎠⎝⎠⎝⎠⎜⎜2cos ⎟ ⎜⎜23⎟⎟ ⎜ e sin xx = ee22 e 2.... That is,
11 1 1 1 −+=−− − .... sin22xx xx x 222 cos 2 42 cos 2 62 cos 2 2223
The last series can be obtained by term by term series- differentiation of 6.3.
sin x 6.5 Product Integration of x
sin x BB B B logdx=−24 (2) x35 + (2) x − 6 (2) x 7 + 8 (2) x 9 − .... ∫ x 4⋅⋅ 3! 8 5! 12 ⋅ 7! 16 ⋅ 9!
Where the BBB246,,, ...are the Bernoulli Numbers.
32 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Proof: Product integrating 6.1, sinx log dx logcosxxdx logcos dx logcos x dx ∫ 24 8 eeeex = ∫∫∫...
By [Grob, p.113, 8b], for x < π , up to a constant,
sinx logdx=− log sin xdx log xdx ∫∫∫x BB B B =−24(2xx )35 + (2 ) − 6 (2 x ) 7 + 8 (2 x ) 9 − .... , 4⋅⋅ 3! 8 5! 12 ⋅ 7! 16 ⋅ 9! where the BBB246,,, ...are the Bernoulli Numbers.
By [Grob, p.113, 9b], for x < π , up to a constant,
log cosxxxdx= 2 log cos d ∫∫222
⎛⎞(2246−−− 1)BBB (2 1) (2 1) ⎜ 246357⎟ =−2⎜ (xxx ) + ( ) − ( ) + ....⎟ ⎝⎠⎜ 43!⋅⋅⋅ 85! 127! ⎟
log cosxxxdx= 4 log cos d ∫∫444
⎛⎞(2246−−− 1)BBBxxx (2 1) (2 1) ⎜ 246357⎟ =−4⎜ ( ) + ( ) − ( ) + ....⎟ ⎝⎠⎜ 43!⋅⋅⋅ 2 85! 2 127! 2 ⎟
…………………………………………………………………………..
Comparing the coefficients of xxx357,,,... on both sides, does not yield any new result.
33 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
sin z 6.6 Euler’s 2nd Product for z For any complex number z
2 z 22zz sinz 4cos− 1 4cos23−− 1 4cos 1 =××33 3 ×... z 33 3
Proof: Using the triple angle formula,
sin 3zz=− 3 sin 4 sin3 z, [Zeid, p.57], we write sinz =− 3 sinzz 4 sin3 33
=−−sinzz 3 4[1 cos2 ] 33()
=−4cos2 zz 1 sin ( 33)
=−4cos22zzz 1 4cos − 1 sin ( 3 )( 3322)
=−××−4cos22zzz 1 ... 4cos 1 sin ( 3 ) ( 33nn)
n 2 z 2 z 3 4cos− 1 4cosn − 1 =××x sinx 33 ... x 3n 33 Therefore, for any complex number z ≠ 0,
zz2 z 2 sin z nn4cos− 1 4cos− 1 33=××... 3 z sin z 33 3n Letting n →∞, 2 z 2 z sin z 4cos− 1 4cos2 − 1 =××33... z 33
34 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Since this holds also for z → 0 , it holds for any complex number z including z = 0.
6.7 Geometric Mean Derivative of Euler’s 2nd Product
sin x Presentation for x
2x 4 sin22xx 4 sin cosx 1 4sin 23 −=−33 − − 3... sin xx 3(4 cos22232xx−−− 1) 3 (4 cos 1) 3 (4 cos x 1) 3 3323
Proof: Geometric Mean Differentiating 6.6,
2 x 2 x sinx 4cos− 1 4cos2 − 1 DD(0)=×× (0)33 D (0) ... x 33
⎛⎞⎛⎞22xx sin x ⎛⎞2 x ⎜⎜4cosDD⎟⎟ 4cos ⎛⎞D 4cosD ⎟ ⎜⎜3323⎟⎟ ⎜ x ⎟ ⎜ 3 ⎟ ⎜⎜⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎜22⎟⎟ ⎜ sin xx⎟ ⎜ 2 ⎟ ⎜⎜4cosxx−− 1⎟⎟ 4cos 1 ⎜⎝⎠⎝⎜⎜⎟−⎟⎟⎟4cos 1 ⎠⎝⎟ ⎜⎜23 ⎠⎝ ⎠ eeeex = 33 3...
⎛⎞⎛⎞22xx ⎛⎞⎜⎜4sin⎟⎟ 4sin ⎜⎜2x ⎟ 23⎟⎟ ⎜ ⎜⎜4sin ⎟ 33⎟⎟ ⎜ ⎜⎜3 ⎟ −−⎟⎟ ⎜ ⎜⎜− ⎟ 22xx⎟⎟ ⎜ 32 ⎜⎜2 x ⎟ 3 (4cos−− 1)⎟⎟ ⎜ 3 (4cos 1) cosx 1 ⎜⎜3(4cos− 1)⎟ ⎟⎟ ⎜ ()− ⎝⎠⎝⎠⎝⎠⎜⎜33⎟ 23⎟⎟ ⎜ 3 eeesin xx= e ...
2x 4sin22xx 4sin cosx 1 4sin 23 −=−33 − − 3... sin xx 3(4cos22232xx−−− 1) 3 (4cos 1) 3 (4cos x 1) 3 3323
35 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
7
Product Calculus of sinx
7.1 Euler’s Product Representation for sinz
For any complex number z ,
⎛⎞⎛⎞⎛⎞22 2 ⎜⎜zz⎟⎟⎟ ⎜ z sinzz=−⎜⎜ 1⎟⎟⎟ 1 − ⎜ 1 − .... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π )
The product converges absolutely in any disk zR< .
Proof: sinz = 2 sinzz cos 22
=⋅2 2 sinzz cos sin z +π 44( 22)
=++22 sinzz sinππ sin z 44222( ) ( )
=+++22 sinzz sinπππ 2 sin z cos z 442( ) ( 4444) ( )
=−−23 sinzz sinzz++2ππ sin cos π 44( ) ( 4) ( 44)
=−−+23 sinzz sinzz++2ππ sin sin ππ 44( ) ( 4) ( 442)
= 23 sinzz sinzz++2ππ sin sin π− 44( ) ( 4) ( 4)
=×27 sinzz sinzz++4ππ sin sin π− 88( ) ( 4) ( 4)
×sinzz++23ππ sin23ππ−−zz sin sin ( 8888) ( ) ( ) ( ) …………………………………………………………………………….
36 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
n n−1 =×221− sinzz sinzz++2 ππ sin sin π− 22nn 22 nn
××sinzz++23ππ sin23ππ−−zz sin sin ... ( 2222nnnn) ( ) ( ) ( )
nn−−11 ...× sinzz+−(2 1)ππ sin (2 −− 1) 22nn Now, sinzzz+++πππ sinπππ−−−zzz= 2sin cos 2 sin cos 22nn( 2 n++11 2 n)( 22 n ++ 11 n)
= 2sinzz++ππ cosππ−−zz 2sin cos ( 22nn++11)( 22 nn ++ 11)
=+sinzz sinππ sin − sin ( 2222nnnn)( )
=−sin22π sin z 22nn And,
sinz +2π sin22ππ−zz=− sin22 sin 22nn 2 n 2 n Therefore,
n ¤ sinz =−× 221− sinzz cos sin 2π sin 2 z 22nn() 2 n 2 n
n−1 ×−××sin222π sinzz ... sin 2(2− 1)π − sin 2 ()22nn() 2 n 2 n
That is,
z n 2n sin z 21n − zz 2π 2 22cossinsin=−×nn() n sin z z 22 2 2n n−1 ×−××sin222π sinzz ... sin 2(2− 1)π − sin 2 ()22nn() 2 n 2 n
Letting z → 0 ,
37 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
n n−1 22n = 21− sin22ππ sin2 ×× ... sin 2(2− 1)π 22nn 2 n
Dividing ¤ by this last equation,
⎛⎞⎛⎞sin22zz sin⎛⎞ sin 2 z⎟ ⎜⎜nn⎟⎟⎜ n⎟ sinz =−−××− 2n sinzz cos⎜⎜ 122⎟⎟ 1 ..⎜ 1 2⎟ nn⎜⎜⎟⎟ n−1 ⎟ 22⎜⎜22ππ⎟⎟2 ⎜ 2 (2− 1)π ⎟ ⎝⎠⎝⎠⎜⎜sinnn⎟⎟ sin ⎜ sin ⎟ 22⎝⎠2n
⎛⎞sinzzzz⎛⎞⎛⎞ sin22 sin⎛⎞ sin 2⎟ ⎜ nnnn⎟⎜⎟⎟ ⎜ ⎜ ⎟ =−−××−z ⎜ 2222⎟cosz ⎜⎜ 1⎟⎟ 1 ..⎜ 1 ⎟ ⎜ ⎟ n ⎜⎜⎟⎟⎜ n −1 ⎟ ⎜ z ⎟⎟⎜2 ⎜⎜22ππ⎟⎟2 2 (2− 1)π ⎟ ⎜⎝⎠n ⎜⎜⎝⎠⎝⎠sinnn⎟⎟ sin ⎜ sin ⎟ 2 22⎝⎠2n
Letting n →∞, for any fixed natural number m we have
22 sin2 zzm⎛⎞ sin 22 ⎛π ⎞ 222nnn⎜⎜⎟⎟zz 11−=−⎜⎜⎟⎟ →− 1 sin222mπ ⎜⎜zm⎟⎟ (mmππ )sin π ( ) 2n ⎝⎠22nn ⎝ ⎠
Consequently, the infinite product
⎛⎞⎛⎞⎛⎞22 2 ⎜⎜zz⎟⎟⎟ ⎜ z z ⎜⎜11−−⎟⎟⎟ ⎜ 1 − .... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π )
Converges to sin z .
The convergence is absolute in any disk zR< , because the infinite series
zz22 z 2 +++... ππ222(2 ) (3 π ) converges absolutely in any disk zR< .
38 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Indeed, in zR< ,
22 2 zz z 2 111⎛⎞ +++=...z ⎜ 1 +++ ...⎟ ππ222(2 ) (3 π ) π 222⎝⎠⎜ 2 3 ⎟
2 2 1 π = z π2 6 1 < R2 . 6
7.2 Geometric Mean Derivative of sinx
12xx 2 2 x cotx = −− − −... x ππ22−−−xxx(2 ) 22 (3 π ) 22
⎛⎞⎛⎞⎛⎞22 2 (0) (0) (0)⎜⎜xx⎟⎟⎟ (0) (0) ⎜x Proof: DxDDsin=−−x ⎜⎜ 1⎟⎟⎟ D 1 D ⎜ 1 − .... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π )
2x −−22xx cosx 1 − 22 22 eeeeesin xxx= π22− (2ππ )−−xx (3 ) ... Thus,
12xx 2 2 x cotx = −− − −... x ππ22−−−xxx(2 ) 22 (3 π ) 22
7.3 The Wallis Product for π
π 224466 = .... 2133557
39 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Proof: Wallis product for π follows from the product formula for sin π , [Bert, p. 424]. 2
1sin= π 2 π ⎛⎞⎛⎞⎛⎞111 =−⎜⎜⎜111....⎟⎟⎟ − − 2 ⎝⎠⎝⎠⎝⎠⎜⎜⎜246222⎟⎟⎟
π ⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞111111 =−+−+−+⎜⎜⎜⎜⎜⎜1⎟⎟⎟⎟⎟⎟ 1 1 1 1 1 ... 2224466⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎜⎜⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟
π 133557 = .... 2224466
40 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
8
Product Calculus of cosx
8.1 Euler’s Product Representation for cosz , For any complex number z ,
⎛⎞⎛⎞⎛⎞222 ⎜⎜⎛⎞222zzz⎟⎟⎟ ⎛ ⎞ ⎜ ⎛ ⎞ cosz = ⎜⎜11−−⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ 1... − ⎜ ⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ ⎜ ⎝⎠⎝⎠⎝⎠⎜⎜⎝⎠πππ⎟⎟⎟ ⎝35 ⎠ ⎜ ⎝ ⎠
The convergence is absolute in any disk zR< .
Proof:
sin 2z cosz = 2sinz
⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞22222 ⎜⎜⎜⎜⎜⎛⎞22222zzzzz⎟⎟⎟⎟⎟ ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ 2z ⎜⎜⎜⎜⎜ 1−−⎜⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ 1 1 − 1 − 1 − ... ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ 1 ⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎜⎜⎜⎜⎜⎝⎠πππππ⎟⎟⎟⎟⎟ ⎝⎠2345 ⎝⎠ ⎝⎠ ⎝⎠ = 2 2 2222 ⎛⎞⎛⎞zz⎛⎞⎛⎞⎛⎞⎛⎞ ⎛ ⎞ ⎛⎞zzz ⎛⎞ ⎛⎞ ⎜ ⎜⎜⎟⎟⎟⎜⎜⎜⎜⎟⎟⎟⎟⎜⎜⎜⎟⎟⎟ z ⎜11−−⎜⎜⎟⎟⎟ ⎟⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟⎟⎟1−−−⎜⎜⎜⎟⎟⎟ 1 1 ... ⎝⎠⎜ ⎝⎠ππ⎟⎝⎠⎝⎠⎝⎠⎝⎠⎜⎜⎜⎜ ⎝2 ⎠⎟⎟⎟⎟⎝⎠345πππ ⎝⎠ ⎝⎠
⎛⎞⎛⎞⎛⎞222 ⎜⎜⎛⎞222zzz⎟⎟⎟ ⎛ ⎞ ⎜ ⎛ ⎞ = ⎜⎜11−−⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ 1... − ⎜ ⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ ⎜ ⎝⎠⎝⎠⎝⎠⎜⎜⎝⎠πππ⎟⎟⎟ ⎝35 ⎠ ⎜ ⎝ ⎠
8.2 Geometric Mean Derivative of cosx
88xx 8 x tanx =+ + + ... ππ22−−−(2)(3)(2)(5)(2)xxx 22 π 22
41 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
⎛⎞⎛⎞⎛⎞222 ⎜⎜⎜⎛⎞222xxx⎟⎟⎟ ⎛⎞ ⎛⎞ Proof: DD(0)cosx = (0)⎜⎜⎜ 1−−−⎜⎜⎜⎟⎟⎟⎟⎟⎟ D (0) 1 D (0) 1 ... ⎜⎜⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎝⎠⎝⎠⎝⎠⎜⎜⎜⎝⎠πππ⎟⎟⎟ ⎝⎠35 ⎝⎠
88xx 8 x sin x −− − − 22 22 22 e cos x = eeππ−−−(2xxx ) (3 ) (2 ) e (5 π ) (2 ) ... Thus,
88xx 8 x tanx =+ + + ... ππ22−−−(2xxx ) (3 ) 22 (2 ) (5 π ) 22 (2 )
42 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
9
Product Calculus of tanx
9.1 Product Representation for tanz
For any complex number z ,
⎛⎞⎛⎞⎛⎞22 2 ⎜⎜zz⎟⎟⎟ ⎜ z z ⎜⎜11−−⎟⎟⎟ ⎜ 1 − ... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π ) tanz = ⎛⎞⎛⎞⎛⎞222 ⎜⎜⎛⎞222zzz⎟⎟⎟ ⎛ ⎞ ⎜ ⎛ ⎞ ⎜⎜11−−⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ 1... − ⎜ ⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ ⎜ ⎜⎜⎝⎠⎝⎠⎝⎠⎝⎠πππ⎟⎟⎟ ⎝35 ⎠ ⎜ ⎝ ⎠
The convergence is absolute in any disk zR< .
Proof: 7.1 and 8.1
9.2 Geometric Mean Derivative of tanx
212xx 8 = −+ sin 2xxππ22−−xx 2(2 ) 2
28xx −+ (2ππ )22−−xx (3 ) 2 (2 ) 2
28xx −+ −... (3ππ )22−−xx (5 ) 2 (2 ) 2
Proof:
43 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
⎛⎞⎛⎞⎛⎞222 ⎜⎜⎛⎞xx⎟⎟⎟ ⎛ ⎞ ⎜ ⎛ x ⎞ DxD(0) (0)⎜⎜11−−⎜⎜⎟⎟⎟⎟⎟⎟ D (0) D (0) ⎜ 1... − ⎜ ⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ ⎜ ⎝⎠⎝⎠⎝⎠⎜⎜⎝⎠πππ⎟⎟⎟ ⎝23 ⎠ ⎜ ⎝ ⎠ Dx(0) tan = ⎛⎞⎛⎞22⎛⎞2 ⎜⎜⎛⎞22xx⎟⎟ ⎛⎞ ⎜ ⎛⎞2x ⎟ DDD(0)⎜⎜11−−⎜⎜⎟⎟⎟⎟ (0) (0) ⎜1...− ⎜ ⎟ ⎟ ⎜⎜⎜⎜⎟⎟⎟⎟⎜ ⎜ ⎟ ⎟ ⎜⎜⎝⎠⎝⎠⎝⎠ππ⎟⎟ ⎝⎠3 ⎝⎠⎜ ⎝⎠5π ⎟
22xx 1 2x −− − 22 22 2 eexxπ22− e(2ππ )−−xx e (3 ) .... e sin 2x = −−88xx − 8 x eeππ22−−−(2xxx ) (3 ) 22 (2 ) e (5 π ) 22 (2 ) Thus,
212xx 8 = −+ sin 2xxππ22−−xx 2(2 ) 2
28xx −+ (2ππ )22−−xx (3 ) 2 (2 ) 2
28xx −+ −... (3ππ )22−−xx (5 ) 2 (2 ) 2
44 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
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Product Calculus of sinh x
10.1 Product Representation of sinh z
⎛⎞⎛⎞⎛⎞22 2 ⎜⎜zz⎟⎟⎟ ⎜ z sinhzz=+⎜⎜ 1⎟⎟⎟ 1 + ⎜ 1 + .... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π )
The convergence is absolute in any disk zR< .
Proof: sinhziiz=− sin , and use 7.1
10.2 Geometric Mean Derivative of sinhx
12xx 2 2 x cothx =+ + + + ... x ππ22+++xxx(2 ) 22 (3 π ) 22
⎛⎞⎛⎞⎛⎞22 2 (0) (0) (0)⎜⎜xx⎟⎟⎟ (0) (0) ⎜ x Proof: DxDxDDsinh=++⎜⎜ 1⎟⎟⎟ 1 D ⎜ 1 + .... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π )
22xx Dxsinh 1 2x 22 22 22 eeeeesinh xx= π +x (2ππ )++xx (3 ) ... Thus,
12xx 2 2 x cothx =+ + + + ... x ππ22+++xxx(2 ) 22 (3 π ) 22
45 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
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Product Calculus of cosh x
11.1 Product Representation of cosh z
⎛⎞⎛⎞⎛⎞222 ⎜⎜⎛⎞222zzz⎟⎟⎟ ⎛ ⎞ ⎜ ⎛ ⎞ coshz =+⎜⎜ 1⎜⎜⎟⎟⎟⎟⎟⎟ 1 + ⎜ 1 + ⎜ .... ⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ ⎜ ⎝⎠⎝⎠⎝⎠⎜⎜⎝⎠πππ⎟⎟⎟ ⎝35 ⎠ ⎜ ⎝ ⎠
The convergence is absolute in any disk zR< .
Proof: coshziz= cos , and apply 8.1
11.2 Geometric Mean Derivative of coshx
88xx 8 x tanhx =+ + + ... ππ22+++(2xxx ) (3 ) 22 (2 ) (5 π ) 22 (2 )
⎛⎞⎛⎞⎛⎞222 ⎜⎜⎜⎛⎞222xxx⎟⎟⎟ ⎛⎞ ⎛⎞ Proof: DxD(0)cosh=+ (0)⎜⎜⎜ 1⎜⎜⎜⎟⎟⎟⎟⎟⎟ D (0) 1 + D (0) 1 + .... ⎜⎜⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎝⎠⎝⎠⎝⎠⎜⎜⎜⎝⎠πππ⎟⎟⎟ ⎝⎠35 ⎝⎠
Dxcosh 88xx 8 x 222222 eeeecoshx = ππ+++(2xxx ) (3 ) (2 ) (5 π ) (2 ) .... Thus,
88xx 8 x tanhx =+ + + ... ππ22+++(2)xxx (3) 22 (2) (5) π 22 (2)
46 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
12
Product Calculus of tanhx
12.1 Product Representation for tanhz
For any complex number z ,
⎛⎞⎛⎞⎛⎞22 2 ⎜⎜zz⎟⎟⎟ ⎜ z z ⎜⎜11++⎟⎟⎟ ⎜ 1 + ... ⎝⎠⎝⎠⎝⎠⎜⎜ππ222⎟⎟⎟(2 ) ⎜ (3 π ) tanhz = ⎛⎞⎛⎞⎛⎞222 ⎜⎜⎛⎞222zzz⎟⎟⎟ ⎛ ⎞ ⎜ ⎛ ⎞ ⎜⎜11++⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ 1... + ⎜ ⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ ⎜ ⎜ ⎝⎠⎝⎠⎝⎠⎜⎜⎝⎠πππ⎟⎟⎟ ⎝35 ⎠ ⎜ ⎝ ⎠
The convergence is absolute in any disk zR< .
Proof: 10.1 and 11.1
12.2 Geometric Mean Derivative of tanhx
212xx 8 = +− sinh2xxππ22++xx 2(2 ) 2
28xx +− (2ππ )22++xx (3 ) 2 (2 ) 2
28xx +− +... (3ππ )22++xx (5 ) 2 (2 ) 2
Proof:
47 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
⎛⎞⎛⎞⎛⎞222 ⎛⎞xx⎟⎟⎟ ⎛ ⎞ ⎛ x ⎞ (0) (0)⎜⎜⎜⎜⎟⎟⎟⎟⎟⎟ (0) (0) ⎜ ⎜ DxD⎜⎜11++⎜⎜⎟⎟⎟⎟⎟⎟ D D ⎜ 1... + ⎜ ⎜⎜⎝⎠⎝⎠⎝⎠⎝⎠πππ⎟⎟⎟ ⎝23 ⎠ ⎜ ⎝ ⎠ D(0) tanhx = ⎛⎞⎛⎞22⎛⎞2 ⎛⎞2x ⎟⎟⎛⎞22xx ⎛⎞⎟ (0)⎜⎜⎜ ⎟ ⎟⎟ (0)⎜⎜⎟ (0) ⎜ ⎟ ⎟ DDD⎜⎜111+++⎜ ⎟ ⎟⎟⎜⎜⎟ ⎜ ⎟ ⎟... ⎜⎜⎝⎠⎝⎠⎝⎠π ⎟⎟⎝⎠35ππ⎝⎠⎜ ⎝⎠⎟
Now,
cosh22xx− sinh Dxtanh2 1 2 ===cosh x . tanhxxxxx sinh sinh cosh sinh2 cosh x Therefore,
22xx 1 2x 22 22 22 2 eex π +x ee(2ππ )++xx (3 ) ... sinh 2x e = 88xx 8 x 22 22 22 eeππ+++(2)(3)(2)(5)(2)xxx e π... Thus,
212xx 8 = +− sinh2xxππ22++xx 2(2 ) 2
28xx +− (2ππ )22++xx (3 ) 2 (2 ) 2
28xx +− +... (3ππ )22++xx (5 ) 2 (2 ) 2
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13 x Product Calculus of e
13.1 Product representation of ex
n ⎛⎞z ⎜1 +→⎟ ez ⎝⎠⎜ n ⎟ n→∞
13.2 Geometric Mean Derivative of ex
De(0) x = e
nn ⎛⎞⎡⎛⎞⎤xx (0)⎜⎜⎟⎟⎢⎥ (0) Proof: DD⎜⎜11+=⎟⎟ + ⎝⎠⎜⎜nn⎟⎟⎣⎦⎢⎥ ⎝⎠
n ⎡ D(1+x ) ⎤ ⎢ n ⎥ 1+x ⎢ n ⎥ = ⎢e ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
⎡ 1 ⎤n ⎢ n ⎥ 1+ x ⎢ n ⎥ = ⎢e ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ 1 1+x n ⎯⎯⎯⎯→e = e n→∞
Thus, De(0) x = e.
49 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
x 13.3 Geometric Mean Derivative of ee
xx De(0) ee= e
nn ⎛⎞⎡⎛⎞⎤eexx Proof: DD(0)⎜⎜11+=⎟⎟⎢ (0) +⎥ ⎜⎜⎟⎟⎢ ⎥ ⎝⎠⎜⎜nn⎟⎟⎣ ⎝⎠⎦
x n ⎡ D(1+e ) ⎤ ⎢ n ⎥ x ⎢ 1+e ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
n ⎡ ex ⎤ ⎢ n ⎥ x ⎢ 1+e ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
ex x 1+e = e n
x → ee n→∞
xx Thus, De(0) ee= e.
k 13.4 Geometric Mean Derivative of ex
kk−1 De(0) xkx= e
50 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
nn ⎛⎞⎡⎛⎞⎤xxkk Proof: DD(0)⎜⎜11+=⎟⎟⎢ (0) +⎥ ⎜⎜⎟⎟⎢ ⎥ ⎝⎠⎜⎜nn⎟⎟⎣ ⎝⎠⎦
k n ⎡ D(1+x ) ⎤ ⎢ n ⎥ k ⎢ 1+x ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
k−1 n ⎡ k x ⎤ ⎢ n ⎥ k ⎢ 1+x ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
kxk−1 k 1+x = e n
k −1 → ekx n→∞
kk−1 Thus, De(0) xkx= e .
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14
Geometric Mean Derivative by
Exponentiation
Geometric Mean Derivative can be obtained by using the Product Calculus of the exponential function. We demonstrate this method by examples.
14.1 Dxe(0)sin = cotx
Proof: Since
sinxe==+log sin xn lim (1log sin x ) , n→∞ n we apply the Geometric Mean Derivative to (1+ log sin x )n . n
⎧⎛ ⎞ ⎛ ⎞⎫ (0) ⎪⎪log sinxx log sin D ⎨⎬⎜⎜1...1+××+=⎟⎟ ⎪⎪⎜⎜nn⎟⎟ ⎩⎭⎪⎪ ⎝⎠⎝⎠ n factors
⎛⎞⎛⎞log sinxx log sin =+DD(0)⎜⎜1⎟⎟ ××+ ... (0) 1 ⎜⎜nn⎟⎟ ⎝⎠⎝⎠ n factors
52 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
n ⎡ D(1+ log sinx ) ⎤ ⎢ n ⎥ ⎢ 1+ log sinx ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
⎡ cotx ⎤n ⎢ n ⎥ ⎢ 1+ log sinx ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ cotx = e1logsin/+ xn
→ ecotx . n→∞
Thus, Dxe(0)sin = cotx .
14.2 Dx(0) x = ex
Proof: Since
xexxx==log lim (1 +xxlog ) n, n→∞ n we apply D(0) to (1+ xxlog )n . n
nn ⎛⎞⎡⎛⎞⎤xxlog xx log (0)⎜⎜⎟⎟⎢⎥ (0) DD⎜⎜11+=+⎟⎟ ⎝⎠⎝⎠⎜⎜nn⎟⎟⎣⎦⎢⎥
53 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
n ⎡ D(1+xxlog ) ⎤ ⎢ n ⎥ ⎢ 1+ xxlog ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥
⎡ 1log+ x ⎤n ⎢ n ⎥ ⎢ 1+xxlog ⎥ = ⎢e n ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ 1log+ x 1+xxlog = e n
→=eex1log+ x n→∞
Thus, Dx(0) x = ex.
54 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
15
Product Calculus of Γ()x
On the half line x > 0, Euler defined the real valued Gamma function by
t =∞ Γ=()xetdt∫ −−tx 1 . t =0 In the half plane Rez > 0, the complex valued integral
t =∞ ∫ et−−tz 1 dt t =0 converges, and is differentiable with
tt=∞ =∞ Detdtettdt−−tz11= −− tz ln z ∫∫. tt==00 Thus, the complex valued integral extends the Euler integral into an analytic function in the half plane Rez > 0. It is denoted by Γ()z .
This function can be further extended to a product representation that is analytic for any z , except for simple poles that it has at z =−−−0, 1, 2, 3, ... The function 1/Γ (z ), that is given by the inverse product, is analytic for any z , with simple zeros at z =−−−0, 1, 2, 3, ...
55 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Therefore, the natural calculus for Γ()z and for 1/Γ (z ) in the complex plane is the product Calculus.
15.1 Euler’s Product Representation for Γ()z
zzz ()1+++111() 1() 1 .... Γ=()z 123 z 111....+++zzz ( 123)( )( ) Proof:
t =∞ Γ=()zetdt∫ −−tz 1 t =0
t =∞ n ⎛⎞t =−lim⎜ 1 ⎟ tdtz −1 ∫ n→∞⎝⎠⎜ n ⎟ t =0 Uniform convergence allows order change of limit, and integration
t =∞ n ⎛⎞t =−lim⎜ 1 ⎟ tdtz −1 n→∞ ∫ ⎝⎠⎜ n ⎟ t =0 The change of variable, utn= / , du= dt/ n , gives
u=1 n =−limnuuduzz() 1 −1 n→∞ ∫ u=0 nz can be written as a product
nz nnzz=+(1) (1)n + z
zzz z =+111...1111 + + + 1nz ()123()()()n (1)n + z
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Integrating by parts with respect to u , keeping z , and n fixed uu==11z nn⎛⎞u ()11−=−uuduz −1 () ud⎜ ⎟ ∫∫⎜ z ⎟ uu==00⎝⎠
u=1 zzu=1 ⎡⎤uunn =−⎢⎥()11udu −() − ⎢⎥zz∫ ⎣⎦u=0 u=0
u=1 1 n−1 =−unz ()1 u du z ∫ u=0
u=1 z +1 nun−1 ⎛⎞ =−()1 ud⎜ ⎟ zz∫ ⎜ + 1⎟ u=0 ⎝⎠
u=1 z +2 nn− 1 n−2 ⎛⎞ u =−()1 ud⎜ ⎟ zz++12∫ ⎜ z ⎟ u=0 ⎝⎠
u=1 z +3 nn−−12 nn−3 ⎛⎞ u =−()1 ud⎜ ⎟ zz++12 z∫ ⎜ z + 3⎟ u=0 ⎝⎠ …………………………………………………….
u=1 zn+ nn−−12 n n −− (1) nnn− ⎛⎞ u =−...() 1 ud⎜ ⎟ zz++12 z z +− n 1∫ ⎜ z + n⎟ u=0 ⎝⎠
u=1 nn−−12 n n −− (1) n⎛⎞ uzn+ = ... ⎜ ⎟ zz++12 z z +− n 1⎜ z + n⎟ ⎝⎠u=0
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nn−−12 n n −− (1)1 n = ... zz++12 z z +−+ n 1 z n
nn−−12 n n −− (1)1 n = ... zz++12 z z +−+ n 1 z n
11 1 1 1 = ... z zz++11 z ++ 11 z ( 12)( ) ( nn− 1)( ) Therefore,
u=1 n limnuuduzz() 1 −=−1 n→∞ ∫ u=0
zzz z 111 1nz =++++×lim( 1) ( 1) ( 1) ...( 1 ) z n→∞{ 123 n (1)n + ⎫ 11 1 1 1⎪ × ... ⎬ z zz++11 z ++ 11 z⎪ ( 12)( ) ( nn− 1)( )⎭⎪
zzz z (111...1+++111) ( ) ( ) ( + 1) = lim 123 n n→∞ z zz++11...11 z ++ z ( 12)( ) ( nn− 1)( )
zzz (1+++111) ( 1) ( 1) .... = 123. z 111....+++zzz ( 123)( )( )
15.2 Geometric Mean Derivative of Γ()x
Γ '(x )⎛⎞ 1 1 1 1 =−−−−−lim⎜ logn ... ⎟ Γ+++()xxxxxnn→∞⎜⎝⎠ 1 2 ⎟
58 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Proof:
xxx DDD(0)(111....+++111) (0)( ) (0) ( ) Dx(0)Γ=() 123 DxD(0) (0)111....+++xxx D (0) D (0) ( 123) ( ) ( )
xx D2x DD(3/2) (4/3) xx ee2x (3/2) e (4/3) .... = 1 111 eex xxx+++123 e e ....
234 .... = 123 1 111 + ++.... ex xxx+++123
1 111 ⎛⎞234 −− −−.... = ⎜ ....⎟e x xxx+++123 ⎝⎠⎜ 123 ⎟
1 11 1 −− −−..... =+lim (ne 1) x xx++12 xn + n→∞
1 11 1 −− −−..... = lim ne x xx++12 xn + n→∞
1 11 1 logn− −−−−.... = lim e x xx++12 xn + n→∞
⎛⎞ ⎜ 11 1 1⎟ lim⎜ logn−− − − ..... − ⎟ = en →∞⎝⎠⎜ xx++12 x xn +⎟
Γ'(x ) Since Dxe(0)Γ=() Γ (x ) ,
59 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Γ'(x )⎛⎞ 1 1 1 1 =−−−−−lim⎜ logn ... ⎟ Γ+++()xxxxxnn→∞⎝⎠⎜ 1 2 ⎟
15.3 Γ=(1) 1 Proof: (1+++111)( 1)( 1) .... Γ=(1)123 = 1 1 1+++111 1 1 .... ( 123)( )( )
15.4 Γ+(1)()zzz =Γ
Proof: zzz+++111 (111....+++111) ( ) ( ) Γ+(1)z = 123 z 1+++zzz+++111 1 1 .... ( 123)( )( )
zzz ()1+++11314 2() 1() 1 .... = 12233 z ++1 1zzz+++111 1 + 1 + .... ()()123()()
zzz ()1+++111() 1() 1 ... = 123 12++++zz12334zz ... ()2322433()()()
zzz (111...+++111) ( ) ( ) = 123 1111...++++z zzz ( )( 234)( )( )
=Γzz().
60 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
15.5 Product Reflection Formula for Γ()z
1 ΓΓ−=()(1zz ) zz1−−−2 1zz22 1 ... ( )( 2322)( )
Proof:
zzz111−−− z z z ()1+++111() 1() 1 ...() 1 + 1() 1 + 1() 1 + 1 ... ΓΓ−=()(1zz ) 123 1 2 3 zz1+++−+++zzz 1 1 ... 1 1111−−− z 1 z 1 z .. ()123()()( )() 1() 2() 3
(111...+++111)( )( ) = 123 zz111..1211+++zz − z − z34 − z 1.. − z ( )( 23)( ) ( ) ( 22334) ( ) ( )
1 = zz111..1111..+++zzz −−−− zzz ()123()()()() 234()()
1 = zz111111..+−+−+− z zzzz ( )( )( 2233)( )( )( )
1 = . zz111...−−−2 zz22 ( )( 2322)( )
π 15.6 ΓΓ−=()(1zz ) sin πz
Proof: By 15.5,
1 ΓΓ−=()(1zz ) zz1−−−2 1zz22 1 ... ()()2322()
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π = 222 πz 111..−−−()πππzzz () () ( πππ222)( (2 ))( (3 ) )
π = . sin πz
2 15.7 Γ=()1 π ( 2 )
π Proof: Substituting z = 1 in ΓΓ−=()(1zz ) , we obtain 2 sin πz π ΓΓ−=()11()1 . 22sin π 2 That is, 2 ⎡⎤Γ=()1 π . ⎣⎢ 2 ⎦⎥
This can be obtained directly through the Wallis Product for π .
2 2 2 4 4 6 6 8 8 10 10 12 12 15.8 ⎡⎤Γ=⋅()1 2... ⎣⎦⎢⎥2 13355779 9 111113
Proof: By 15.5,
2 1 ⎡⎤Γ=()1 ⎣⎦⎢⎥2 11⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞ 1 1 1 1 ⎜⎜⎜⎜⎜1−−−−−⎟⎟⎟⎟ 1 1 1 1 ⎟ ... 2⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎜⎜⎜⎜⎜2468102222⎟⎟⎟⎟ 2 ⎟
1 = 113355779911⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞⋅⋅⋅⋅⋅ ⎜⎜⎜⎜⎜⎟⎟⎟⎟ ⎟... 2⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎜⎜⎜⎜⎜2468102222⎟⎟⎟⎟ 2 ⎟
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⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞22 44 66 88 1010 =⋅2....⎜⎜⎜⎜⎜ ⋅⎟⎟⎟⎟ ⋅ ⋅ ⋅ ⋅ ⎟ ⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎜⎜⎜⎜⎜13⎟⎟⎟⎟ 35 57 79 9 11 ⎟
Wallis Formula of 7.3, follows from 15.7, and 15.8.
63 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
16
Products of Γ()z
Γ+(1z1 ) 16.1 , wherezww112=+ Γ+(1ww12 ) Γ+ (1 )
ww ww 11++12 11 ++ 12 Γ+(1z ) ()()11++ww()22()() 33() 1 =×××12 ... zz Γ+(1ww12 ) Γ+ (1 )() 1 + z 1 11++11 ()23()
Proof:
Γ+(1z ) 1 = Γ+(1ww12 ) Γ+ (1 )
111+++zzz (111....+++111) 111( ) ( ) =×123 111+++zzz 1111....++z 111 + + ()1 ( 123)( )( )
111+++www111 (1111...++w1 )( )( +)( +) ××123 111+++www 111...+++111111 ( 123) ( ) ( )
111+++www222 (11++w2 )( )( 1 +)( 1 +) ... × 123. 111+++www 111...+++111222 ( 123) ( ) ( )
zzz (1+++111) 111( 1) ( 1) .... =×123 zzz 1111....++++z 111 ()1 ( 234)( )( )
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www111 ()1111...++++w1 ()()() ××234 www 111...+++111111 ()123()()
www222 ()1+w2 () 111... +++()() × 234. www 1+++111222 1 1 ... ()123()()
ww11 ww 22 (1+++ww12)( 1)( 1) ...( 1 +++)( 1)( 1) ... = 23 23 zzz 1111...++++z 111 ()1 ( 234)( )( )
ww ww 11++12 11 ++ 12 ()()11++ww( 22)( ) ( 33)( ) =×××12 ... zz ()1 + z1 11++11 ( 23) ( )
Γ(1) 16.2 = sinh x Γ+(1ix ) Γ− (1 ix )
Proof:
Γ(1) =+−()()1ix 11111... ix () +−+−ix() ix() ix() ix Γ+(1ix ) Γ− (1 ix ) 2233
=+111...x 2 +xx22 + ( )( 2322)( ) = sinh x .
Similarly, we obtain
65 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
Γ+(1zz12 ) Γ+ (1 ) 16.3 , where zz12+= www 1 + 2 + 3. Γ+(1www123 ) Γ+ (1 ) Γ+ (1 )
Γ+(1zz ) Γ+ (1 ) 12= Γ+(1www123 ) Γ+ (1 ) Γ+ (1 )
www 111+++123 ()()()111+++www( 222)( )( ) =××123 zz ()()11++zz12 11++12 ( 22)( )
www (111+++123)( )( ) ××333... zz 11++12 ( 33)( )
More generally,
16.4 If zz12+++=... zkl ww 1 + 2 ++ ... w,
Γ+(1zz ) Γ+ (1 )... Γ+ (1 z ) 12 k may be simplified Γ+(1ww12 ) Γ+ (1 ).... Γ+ (1 wl )
A weaker result that requires that kl= , is stated in [Melz, p. 101], and in [Rain, p. 249].
66 Gauge Institute Journal, Volume 4, No 4, November 2008 H. Vic Dannon
17
Product Calculus of Jxν()
For a complex number ν , the Bessel function Jzν () solves Bessel’s differential equation
dw2 dw zzzw222++−=()0ν . dz 2 dz
For ν real, Jzν () has infinitely many real zeros, all simple with the possible exception of z = 0.
For ν ≥ 0 , the positive zeros jν,k are a monotonic increasing sequence
jjjννν,1<<< ,2 ,3 ...
17.1 Product Formula for Jzν () [Abram, p.370]
ν ⎛⎞⎛⎞⎛⎞ ⎛⎞zzzz1 ⎜⎜⎜222⎟⎟⎟ Jz()=−−−⎜ ⎟ ⎜⎜⎜ 1⎟⎟⎟ 1 1 ... ν ⎝⎠⎜ 2(1)⎟ Γ+ν ⎜⎜⎜222⎟⎟⎟ ⎝⎠⎝⎠⎝⎠⎜⎜⎜jjjννν,1⎟⎟⎟ ,2 ,3
17.2 Geometric Mean Derivative of Jzν ()
DJ() x ν 222xxx ν =− − − −... Jx() x 222222 ν jxjxjxννν,1−−− ,2 ,3
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1 Proof: DJx(0)()=× D (0) Dx (0) ν ν 2(ν Γ+ν 1)
⎛⎞⎛⎞⎛⎞ ⎜⎜⎜xxx222⎟⎟⎟ ×−DDD(0)⎜⎜⎜1⎟⎟⎟ (0) 1 − (0) 1 − ... ⎜⎜⎜222⎟⎟⎟ ⎝⎠⎝⎠⎝⎠⎜⎜⎜jjjννν,1⎟⎟⎟ ,2 ,3
DxjDxjDxj(1−−−22 / ) (1 22 / ) (1 22 / ) Dxν ννν,1 ,2 ,3 1/−−−xj22 1/ xj 22 1/ xj 22 = ee0 xν eννν,1 e ,2 e ,3 ...
−−−222xxx ν jxjxjx222222−−− = eex ννν,1 e ,2 e ,3 ...
222xxx −−− ν jxjxjx222222−−− = eex ννν,1 ,1 ,2 ... Thus,
DJ() x ν 222xxx ν =− − − −... Jx() x 222222 ν jxjxjxννν,1−−− ,2 ,3
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Product Calculus of Trigonometric
Series
If 1 ∞ ⎛⎞nxππ nx fx()=+ a⎜ a cos + b sin ⎟ 0 ∑⎜ nn⎟ 2 n=1⎝⎠LL the Geometric Mean Derivative can be applied to
⎛⎞⎛ ⎞⎛ ⎞ ⎜⎜122⎟⎟⎟ππxx ⎜ π x π x ⎜⎜fx( )−+ a01122⎟⎟⎟ a cos b sin ⎜ a cos + b sin e⎝⎠⎝⎜⎜2 ⎟⎟⎟= eeLL ⎠⎝ ⎜ L L ⎠....
18.1 Product Integral of a Trigonometric Series on [0,2π ], ⎛⎞sinxxx sin2 sin 3 x =−π 2...⎜ + + +⎟ ⎝⎠⎜ 12 3⎟ Therefore, sin 2xx sin 3 −−22 eeeexx−−π = 2sin 23..... Product Integrating both sides,
⎛⎞ ⎛⎞⎛⎞ ⎜⎜⎜12 ⎟⎟⎟ cos 2xx cos 3 ⎜⎜⎜xx−π ⎟⎟⎟22 eeee⎝⎠⎜⎜⎜2 ⎟⎟⎟= 2cosx ⎝⎠⎝⎠2322.... Hence,
1 cos2xx cos 3 xx2 −=π 2cos22.... x + + + 2 2322
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Infinite Functional Products
Euler represented Analytic functions by infinite products, [Saks]. to which the Geometric Mean derivative may be applied.
19.1 Geometric Mean Derivative of an Euler Product
Consider Euler’s product
1 23 =+()1xx() 1 +22 1 + x 1 + x 2 ..... 1 − x ()() Applying the Geometric Mean Derivative to both sides,
23 n ⎛⎞ 21− 2222xx−− 1 32 1 2n x 2 − 1 ⎜ 1 ⎟ 1 2x ⎜ ⎟ 23 n eeeeee⎝⎠⎜1−x ⎟ = 11++xx22 1 + x 1 + x 2.... 1 + x 2 ... Hence,
12 n 112212xx−− 1 22 1 2n x 2 − 1 =+12 + +...n + ... 11−+xx11++xx22 1 + x 2
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Path Product Integral
20.1 Path Product Integral in the plane
Let Pxy(, ), and Qxy(, ) be positive, and smooth so that
logPxy ( , ) is integrable with respect to x ,
and logQxy ( , ) is integrable with respect to y , along the path γ in the xy, Plane, from (,xy11 ) to (,xy22 ). We define the Product Integral along the path γ by
dx dy dx dy ∏∏∏(Pxy(, )) ( Qxy (, )) ≡ ( Pxy (, )) ( Qxy (, )) γγγ
∫ ()()log(,)Pxy dx∫ log(,) Qxy dy = eeγγ
∫ ()()logPxy ( , ) dx+ log Qxy ( , ) dy = e γ
20.2 Green’s Theorem for the Path Product Integral
Path Product Integral over a loop equals
⎛⎞∂ P ∂ Q ⎜ y x ⎟ ∫∫ ⎜ − ⎟dxdy ⎝⎠⎜ PQ⎟ einerior()γ Proof: By Green’s Theorem.
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20.3 Path Product Integral in E 3
If γ is a path on a smooth surface in three dimensional space, we define the path product Integral along γ by
()()()logPxyz ( , , ) dx++ log Qxyz ( , , ) dy log Rxyz ( , , ) dz dx dy dz ∫ ∏()()()Pxyz(,,) Qxyz (,,) Rxyz (,,) ≡ eγ γ
20.4 Stokes Theorem for the Path Product Integral path product Integral over a loop in a smooth surface in E 3 equals ⎡ ⎤⎛ ⎞ ⎢⎥logP ( x , y , z ) ⎜ dydz ⎟ ⎜ ⎟ ∇×⎢⎥logQ ( x , y , z ) i⎜ dxdz ⎟ ∫∫ ⎢⎥⎜ ⎟ ⎢⎥⎜ ⎟ ⎢⎥logR ( x , y , z ) ⎝⎠⎜ dxdy ⎟ e ⎣⎦.
Proof: By Stokes Theorem.
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Iterative Product Integral
21.1 Iterative Product Integral of f (,)xt
Let f (,)xt be positive in the rectangle
[,][,]xx01× tt 01 so that logf (xt , ) is integrable on the rectangle. Then, the double product integral is defined iteratively by
dt dt ⎛⎞xx= 1 tt= xx= tt= 1 ⎛⎞1 1 ⎜ ∫ logfxtdx ( , ) ⎟ ⎜ dx ⎟ ⎜ xx= ⎟ ⎜ fxt(,) ⎟ = ⎜ e 0 ⎟ ∏⎜ ∏ ⎟ ∏⎜ ⎟ tt= ⎝⎠⎜ xx= ⎟ tt= ⎜ ⎟ 0 0 0 ⎝⎠⎜ ⎟
ttxx==11 ∫∫logfxtdxdt ( , ) = ettxx==00
21.2 Iterative Product Integral of erxtdx(,)
ttxx==11 dt rxtdxdt(,) tt==11⎛⎞ xx ∫∫ ⎜ ⎟ ttxx== ⎜ eerxtdx(,) ⎟ = 00 ∏∏⎜ ⎟ tt==00⎝⎠ xx
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Harmonic Mean Integral
22.1 Harmonic Mean Integral
The electrical voltage on a capacitance Cq()due to a charge dq at x is dq dq() x dx =≡, Cq() Cqx (()) fx () where the function f()x is real and non-vanishing.
ba− Over n equal sub-intervals of the interval [,]ab, Δ=x , n we obtain the sequence of voltages
11 1⎛⎞ 111 Δ+xx Δ+... + Δ= x⎜ + + ... +⎟ Δ x. ⎜ ⎟ fx()12 fx () fx (nn )⎝⎠ fx () 12 fx () fx ( ) As n →∞, the sequence converges to
xb= 1 dx . ∫ fx() xa= We call the limit the Harmonic Mean integral of f()x over the interval [,]ab, and denote it by xb= Ifx(1)− (). xa= The inverse operation to Harmonic Mean integration is the Harmonic Mean Derivative
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Harmonic Mean, and Harmonic Mean
Derivative
23.1 The Harmonic Mean of f()x over [,]ab
Given an non-vanishing integrable function f()x over the interval
[,]ab, partition the interval into n sub-intervals, of equal length
ba− Δ=x , n choose in each subinterval a point
ci , and consider the Harmonic Mean of f()x
1 ba− n = 11 1⎛⎞11 1 +++... ⎜ +++Δ... ⎟ x ⎜ ⎟ fc()12 fc () fc (n ) ⎝⎠fc()12 fc () fc (n )
As n →∞, the sequence of Harmonic Means converges to
ba−− ba = , xb= 1 Hb()− Ha () dx ∫ fx() xa= where
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tx= 1 Hx()= dt. ∫ f ()t t =0 Therefore, ba−
xb= 1 dx ∫ fx() xa= is defined as the Harmonic Mean of f()x over [,]ab.
23.2 Mean Value Theorem for the Harmonic Mean
ba− There is a point acb<<, so that = f ()c xb= 1 dx ∫ fx() xa=
23.3 The Harmonic Mean of f ()x over [,xx+ dx ]
The Harmonic Mean at x is the Inverse operation to Harmonic Mean Integration Proof: The Harmonic Mean of f ()x over the interval [,xx+Δ x ], is
Δx . tx=+Δ x 1 dt ∫ f ()t tx= Letting Δ→x 0, the Harmonic Mean of f ()x at x equals f ()x .
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Δx lim= f (x ). tx=+Δ x Δ→x 0 1 dt ∫ ft() tx= Thus, the operation of finding the Harmonic Mean of f()x at the point x , is inverse to Harmonic Mean Integration.
This leads to the definition of the Harmonic Mean Derivative
23.4 Harmonic Mean Derivative
The Harmonic Mean Derivative of
tx= 1 Hx()= dt ∫ f ()t t =0 at x is defined as the Harmonic Mean of f ()x at x
Δx DHx(1)− ()≡ lim tx=+Δ x Δ→x 0 1 dt ∫ ft() tx=
1 23.5 DHx(1)− ()= DHxx ()
Proof: dx DHx(1)− ( )= Standard Part of Hx()()+− dx Hx 1 = . DHxx ()
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Harmonic Mean Calculus
24.1 The Fundamental Theorem of the Harmonic Mean
Calculus
⎛⎞tx= ⎜ ⎟ DIftfx(1)(1)−−⎜ ()⎟ = ( ). x ⎜ ⎟ ⎝⎠⎜ ta= ⎟
⎛⎞tx= ⎛⎞tx= ⎟ ⎜ ⎟ (1)(1)−−⎜ ⎟ (1) −⎜ 1 ⎟ Proof: DIftDxx⎜ ()⎟ = ⎜ dt⎟ ⎜ ta= ⎟ ⎜ ∫ ft() ⎟ ⎝⎠⎜ ⎟ ⎝⎠⎜ta= ⎟
1 = tx= 1 Ddt x ∫ f ()t ta=
= f ()x .
24.2 Table of Harmonic Mean Derivatives and integrals
We list some Harmonic Mean Derivatives, and Integrals.
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f()xfxIDfx(1)− () (1)− () xx1log x 2 11− 2xx xa 11 axaa−−11(1) a− x 11 −x 22x x 2 1 logxx dx ∫ logx eexx−−− e x 11 e22xxee−−− 2 x 22 11 x x dx xxxx(log+ 1) ∫ x sinx 1 dx cosxx∫ sin cosx 1 dx sinxx∫ cos tanxx sin2 x logsin xx1 eeeee−x − dx ∫ eex exesinxxcos sin ∫ e−sinxdx execosxx−sin cos ∫ edx−cosx
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Quadratic Mean Integral
25.1 Quadratic Mean Integral
Given a Riemann integrable, positive f()x on [,]ab, partition the interval into n sub-intervals, of equal length
ba− Δ=x , n choose in each subinterval a point
ci , and consider the finite products,
22 2 ()f ()cfc12+++Δ () ... fcx (n )
As n →∞, the sequence converges to
xb= ∫ f 2()xdx. xa= We call this limit the Quadratic Mean Integral of f()x over the interval [,]ab, and denote it by xb= (2) 2 Ifx()= f 2 xa= Lab[,]
Thus, Quadratic Mean integration transforms a function to its L2 norm squared.
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25.2 Cauchy-Schwartz inequality for Quadratic Mean Integrals 111 ⎛⎞⎛⎞⎛⎞xb===222 xb xb ⎜⎜⎜⎟⎟⎟ ⎜⎜⎜Ifxgx(2)⎡⎤()+≤ ()⎟⎟⎟ Ifx (2) () + Igx (2) () ⎜⎜⎜⎣⎦⎟⎟⎟ ⎝⎠⎝⎠⎝⎠⎜⎜⎜xa===⎟⎟⎟ xa xa Proof: 1 1 ⎛⎞xb= 2 ⎛⎞xb= 2 ⎟ ⎜ 2 ⎟ ⎜ (2) ⎡⎤⎟ ⎜ ⎟ ⎜Ifxgx()+= ()⎟ ⎜ () fxgxdx () + () ⎟ ⎜ xa= ⎣⎦⎟ ⎜ ∫ ⎟ ⎜⎝⎠⎟ ⎜⎝⎠xa= ⎟
By Cauchy-Schwartz Inequality,
1/2 1/2 ⎛⎞⎛⎞xb== xb ⎜⎜22⎟⎟ ≤+⎜⎜()f() x dx⎟⎟ () g () x dx ⎜⎜∫∫⎟⎟ ⎝⎠⎝⎠⎜⎜xa==⎟⎟ xa
11 ⎛⎞⎛⎞xb==22 xb ⎜⎜⎟⎟ =+⎜⎜Ifx(2)()⎟⎟ Igx (2) () . ⎜⎜⎟⎟ ⎝⎠⎝⎠⎜⎜xa==⎟⎟ xa
25.3 Holder Inequality for Quadratic Mean Integrals
11 xb= ⎛⎞⎛⎞xb==22 xb ⎜⎜(2)⎟⎟ (2) f ()()xgx dx≤ ⎜⎜ I fx ()⎟⎟ I gx () ∫ ⎜⎜xa==⎟⎟ xa xa= ⎝⎠⎝⎠⎜⎜⎟⎟
Proof: By Holder’s inequality,
1/2 1/2 xb===⎛⎞⎛⎞ xb xb ⎜⎜22⎟⎟ f()() x g x dx≤ ⎜⎜ f () x dx⎟⎟ g () x dx ∫∫∫⎜⎜⎟⎟ xa===⎝⎠⎝⎠⎜⎜ xa⎟⎟ xa
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11 ⎛⎞⎛⎞xb==22 xb ⎜⎜⎟⎟ ≤ ⎜⎜IfxIgx(2)()⎟⎟ (2) () . ⎜⎜⎟⎟ ⎝⎠⎝⎠⎜⎜xa==⎟⎟ xa
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Quadratic Mean and Quadratic Mean
Derivative
26.1 Quadratic Mean of f()x over [,]ab
Given an integrable positive function f()x on [,]ab, partition the
ba− interval into n sub-intervals, of equal length Δ=x , n choose in each subinterval a point ci , and consider the Quadratic Means of f ()x
1/2 ⎛⎞fc22( )+++ fc ( ) ... fc 2 ( ) ⎛⎞1/2 ⎜ 12 n ⎟ ⎜ 1 22 2 ⎟ ⎜ ⎟ =+++Δ⎜ ()fc(12 ) fc ( ) ... fc (n ) x⎟ ⎝⎠⎜ nba⎟ ⎝⎠⎜ − ⎟
As n →∞, the sequence of Quadratic Means converges to
1/2 ⎛⎞xb= ⎛⎞1/2 ⎜ 1()()2 ⎟ Qb− Qa ⎜ fxdx() ⎟ = ⎜ ⎟ , ⎜ba−−∫ ⎟ ⎜⎝⎠ ba ⎟ ⎝⎠⎜ xa= ⎟ where tx= Qx()= ∫ f2 () tdt. t =0
1/2 ⎛⎞xb= ⎜ 1 2 ⎟ ⎜ fxdx() ⎟ ⎜ba− ∫ ⎟ ⎝⎠⎜ xa= ⎟ is defined as the Quadratic Mean of f()x over [,]ab.
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26.2 Mean Value theorem for the Quadratic Mean
1 ⎛⎞xb= 2 ⎜ 2 ⎟ There is a point acb<<, so that ⎜ 1 f ()xdx⎟ = fc () ⎜ ba− ∫ ⎟ ⎝⎠⎜ xa= ⎟
26.3 The Quadratic Mean of f ()x over [,xx+ dx ]
The Quadratic Mean at x , is the Inverse operation to Quadratic Mean Integration Proof: By 26.2, there is xcx<<+Δ x, so that 1/2 ⎛⎞tx=+Δ x ⎜ 2 ⎟ ⎜ 1 f ()tdt⎟ = fc (). ⎜ Δx ∫ ⎟ ⎝⎠⎜ tx= ⎟
Letting Δ→x 0, the Quadratic Mean of f ()x at x equals f ()x .
1/2 ⎛⎞tx=+Δ x ⎟ ⎜ 1 2 ⎟ lim⎜ f (tdt )⎟ = f ( x ). Δ→x 0⎜ Δx ∫ ⎟ ⎝⎠⎜ tx= ⎟
Thus, the operation of finding the Quadratic Mean of f()x at the point x , is inverse to Quadratic Mean Integration.
This leads to the definition of Quadratic Mean Derivative
26.4 Quadratic Mean Derivative
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The Quadratic Mean Derivative of
tx= Qx()= ∫ f2 () tdt t =0 at x is defined as the Quadratic Mean of f ()x at x
1/2 ⎛⎞tx=+Δ x ⎟ (2)⎜ 1 2 ⎟ DQx()≡ lim⎜ ftdt () ⎟ Δ→x 0⎜ Δx ∫ ⎟ ⎝⎠⎜ tx= ⎟
(2) 1/2 26.5 DQx()= ( DQxx ())
Proof:
1/2 ⎛⎞Qx()()+− dx Qx DQx(2) ( )= Standard Part of ⎜ ⎟ ⎝⎠⎜ dx ⎟
1/2 = (DQxx ())
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Quadratic Mean Calculus
27.1 The Fundamental Theorem of the Quadratic Mean Calculus ⎛⎞tx= ⎜ ⎟ DIftfx(2)⎜ (2) ()⎟ = ( ) x ⎜ ⎟ ⎝⎠⎜ ta= ⎟
⎛⎞tx= ⎛⎞tx= ⎟ ⎜ 2 ⎟ (2)⎜ (2)⎟ (2) ⎜ ⎟ Proof: DIftDxx⎜ ()⎟ = ⎜ () ftdt () ⎟ ⎜ ta= ⎟ ⎜ ∫ ⎟ ⎝⎠⎜ ⎟ ⎝⎠⎜ ta= ⎟
1 ⎡ ⎛⎞⎤tx= 2 ⎢ ⎜ 2 ⎟⎥ = Dftdt⎜ () ⎟ ⎢ x ⎜ ∫ ()⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝⎠ta= ⎟⎦
= f ()x .
27.2 Table of Quadratic Mean Derivatives and integrals
We list some Quadratic Mean Derivatives, and Integrals.
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fx()D(2) fx () I (2)f()x 10 x 20 4x a 0 ax2 1 x 1 x 3 3 1 x2 2 xx5 5 1 xa axaa−+121 x 21a + xx−1 −−11− x −1/2 2 lnx xxdx∫ ()ln 1 eexx/2e 2 x 2 1 ee24xx2 e x 4 1 eeaxa ax/2 e2 ax 2a sinx cosxxdx∫ () sin2
cosx sinxxdx∫ () cos2 1 tanx ()tan2 xdx sin x ∫ 1/2 eesinx ()cosxe()sinx /2 ∫ 2sinxdx 1/2 eecosx ()−sin xedx()cosx /2 ∫ 2cosx
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References
[Abram]. Abramowitz, Milton and Stegun, Irene, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, United States Department of Commerce, National Bureau of Standards, 1964. [Bert] Bertrand, Joseph, “Traite de Calcul Differentiel et de Calcul Integral, Volume I, Calcul Differentiel”, Gauthier-Villars, 1864. Reproduced by Editions Jacques Gabai, 2007. [Doll]. Dollard, John, and Friedman Charles, “Product Integration with Applications to Differential Equations”. Addison Wesley, 1979. [Euler1] Euler, Leonhard, “Introduction to Analysis of the Infinite”, Book I, Springer- Verlag, 1988. [Grob] Grobner, und Hofreiter, “Integraltafel”, Volume I, Springer Verlag, 1975. [Haan] De Haan, D. Bierens, “Nouvelles Tables D’Integrales Definies”, Edition of 1867 – Corrected. Hafner Publishing. [Kaza] Kazarinoff, Nicholas, Analytic Inequalities, Holt, Reinhart, and Winston, 1961. [Melz] Melzak, Z., “Companion to Concrete Mathematics”, Wiley, 1973. [Rain] Rainville, Earl, “Infinite Series”, Macmillan, 1967. [Saks] Saks, Stanislaw, and Zygmond, Antoni, Analytic Functions, Third edition, (Second is fine), Elsevier, 1971. [Spieg] Spiegel, Murray, Mathematical Handbook of Formulas and Tables, McGraw-Hill 1968. [Spiv] Spivey Michael, The Product Calculus, in ABSTRACTS of papers presented to the American Mathematical Society, Volume 28, Number 1, Issue 147, p. 327. [Zeid]. Zeidler, Eberhard, “Oxford User’s Guide to Mathematics”, Oxford University Press, 2004.
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