Delta Function, and Hilbert Transform

Gauge Institute Journal H. Vic Dannon

Delta Function and Hilbert Transform H. Vic Dannon [email protected] May, 2011

Abstract The Hilbert Integral Theorem guarantees that the

Hilbert Transform and its Inverse are well defined operations, so that inversion yields the original function that generated the Transform. It is Fundamental to the theory of the Hilbert Transform, but has not been obtained until now. We establish it here for any hyper-real function in Infinitesimal Calculus, from the properties of the Delta Function. Nor were Conditions for the Existence of the Hilbert Transform established. In general, it may seem that the existence of the Hilbert Transform requires an analytic function, but conditions have been given for functions that are absolutely integrable. That

p is, functions with ∫ gd()ξξ<∞. We use the Delta Function, in Infinitesimal Calculus to establish that any hyper-real function has a Hilbert Transform. The Transform gives rise to two Delta Functions, known in optical

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coherence, δ+()x , and δ−()x . We derive new formulas for these functions.

Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real,

Cardinal, Infinity. Non-Archimedean, Non-Standard Analysis, Calculus, Limit, Continuity, Derivative, Integral, Delta Function, Fourier Transform, Hilbert Transform, Filter,

2000 Mathematics Subject Classification 26E35; 26E30;

26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30; 46F12; 46F25; 46F30; 32A55; 32A45;

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Introduction

0.1 Transfer Function

A Filter composed of a Resistance R , and an Inductance L in series, driven by a harmonic signal at frequency ω , has the impedance Ri+ ωL, and the Transfer Function

1 RL−ω =+i . 222222 RiL+ ω RLRL ++ωω uv(ωω ,0) ( ,0)

The functions u(,0)ω , and v(,0)ω can be extended to the analytic functions R uz()= , RzL22+ 2 and −zL vz()= . RzL22+ 2 It follows that

ξ=∞ u()ξ Cauchy Principal Value of dvξπω= ( , 0) . ∫ ξω− ξ=−∞

To see that, note that on the contour γ

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uz() R 1 >>dz = dz ∫∫z − ω L2 ()()()zi−+−RR zi zω γγLL ⎧⎫ R ⎪⎪1 = 2Resπi ⎨⎬⎪⎪ L2 ⎪⎪()()()zi−+−RR zi zω ⎩⎭⎪⎪LL zi= R L ⎧⎫ R ⎪⎪1 = 2πi lim ⎪⎪⎨⎬ L2 zi→ R ⎪⎪()(zi+−R zω) L ⎩⎭⎪⎪L R 1 = 2πi L2 2(iiRR− ω) LL 1 = π iR− ω L −−iRππω L = . RL222+ ω On the semicircle in the upper-half plane

uz() dz → 0 , as z →∞. ∫ z − ω On the semi-circle that bypasses ω

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uz() R 1 dz = dz ∫∫??z − ω L2 ()()()zi−+−RR zi zω LL ⎧⎫ R ⎪⎪1 =−()Resπ i ⎨⎬⎪⎪ L2 ⎪⎪()()()zi−+−RR zi zω ⎩⎭⎪⎪LL z =ω ⎧⎫ R ⎪⎪1 =− πi lim ⎨⎬⎪⎪ L2 z→ω ⎪⎪()(zi−+RR zi) ⎩⎭⎪⎪LL R 1 =− πi L2 ()(ωω−+iiRR) LL −iRπ = . RL22+ ω 2 Therefore,

ξ=∞ uiRLi()ξππωπ−− −R Principal Value of dξ =− ∫ ξω− 222222 ξ=−∞ RLRL++ωω

−ωL = π RL22+ ω 2 That is, ⎧⎫ξ=∞ 1(⎪⎪u ξ,0) vd(ωξ , 0)= ⎨⎬⎪⎪ princial value of πξ⎪⎪∫ − ω ⎩⎭⎪⎪ξ=−∞

⎧⎫ξωο=− ξ=∞ 1⎪⎪uu (ξξ ,0) ( ,0) =+⎨⎬⎪⎪ddξξ, πξωξω⎪⎪∫∫−− ⎩⎭⎪⎪ξξ=−∞ =ω +ο where ο is an infinitesimal.

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Similarly, it can be shown that

⎧⎫ξ=∞ 1(⎪⎪v ξ,0) ud(ωξ , 0)=− ⎨⎬⎪⎪ princial value of πξ⎪⎪∫ − ω ⎩⎭⎪⎪ξ=−∞

⎧⎫ξωο=− ξ=∞ 1(,0)(,0)⎪⎪vvξξ =− ⎨⎬⎪⎪ddξξ+ , πξωξω⎪⎪∫∫−− ⎩⎭⎪⎪ξξ=−∞ =ω +ο where ο is an infinitesimal.

0.2 Hilbert Transform v(,0)ω is the Hilbert Transform of u(,0)ω

vu(,0)ω = H , and u(,0)ω is the Inverse Hilbert Transform of v(,0)ω

uv(,0)ω = H−1 .

In general, let f()zuxyivxy=+ (, ) (, ) be analytic in y < 0, and consider the contour

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By Cauchy Theorem, f ()z dz = 0 . ∫? zx− γ 0

On the semi-circle that bypasses x0 ,

fz() ⎪⎪⎧⎫fz() dz= iπRes⎨⎬⎪⎪ ∫> ⎪⎪ zx−−00zx ⎩⎭⎪⎪zx= 0

==ifzifxππlim ( ) (0 ) zx→ 0 If on the semi-circle in the lower half plane

f ()z ∫ dz → 0 , as z →∞. zx− 0 then ξ=∞ f ()ξ princial value of difxξπ+=( ) 0. ∫ ξ − x 0 ξ=−∞ 0 That is,

ξ=∞ uiv(,0)ξξ+ (,0) p.v. diuxivxξπ++()( , 0) ( , 0)= 0 ∫ ξ − x 00 ξ=−∞ 0

Therefore, ξ=∞ 1(,0)u ξ vx(,0)== pv .. dξ H u 0 πξ∫ − x ξ=−∞ 0

ξ=∞ 1(,0)v ξ ux(,0)=− pv .. dξ =H−1 v. 0 πξ∫ − x ξ=−∞ 0

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0.3 Existence of the Hilbert Transform

In general, it may seem that the existence of the Hilbert Transform requires an analytic function, but conditions have been given for functions that are absolutely integrable. That is,

p functions with ∫ gd()ξξ<∞. We use the Delta Function, in Infinitesimal Calculus to establish that any hyper-real function has a Hilbert Transform.

0.4 The Hilbert Integral Theorem

The Hilbert Integral Theorem guarantees that the Hilbert Transform and its Inverse are well defined operations, so that inversion yields the original function that generated the Transform. We shall establish it for any hyper-real function in Infinitesimal Calculus, from the properties of the Delta Function.

0.5 Delta functions and the Hilbert Transform

The Hilbert transform gives rise to two Delta Functions, known in optical coherence, δ+()x , and δ−()x . We derive new formulas for these Delta Functions.

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Hyper-real Line

Each real number α can be represented by a Cauchy sequence of rational numbers, (,rrr123, ,...) so that rn → α . The constant sequence (ααα, , ,...) is a constant hyper-real.

In [Dan2] we established that, 1. Any totally ordered set of positive, monotonically decreasing

to zero sequences (ιιι123 , , ,...) constitutes a family of infinitesimal hyper-reals. 2. The infinitesimals are smaller than any real number, yet strictly greater than zero. 3. Their reciprocals (111,,,...) are the infinite hyper-reals. ιιι123 4. The infinite hyper-reals are greater than any real number, yet strictly smaller than infinity. 5. The infinite hyper-reals with negative signs are smaller than any real number, yet strictly greater than −∞. 6. The sum of a real number with an infinitesimal is a non-constant hyper-real. 7. The Hyper-reals are the totality of constant hyper-reals, a family of infinitesimals, a family of infinitesimals with

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negative sign, a family of infinite hyper-reals, a family of infinite hyper-reals with negative sign, and non-constant hyper-reals. 8. The hyper-reals are totally ordered, and aligned along a line: the Hyper-real Line. 9. That line includes the real numbers separated by the non- constant hyper-reals. Each real number is the center of an interval of hyper-reals, that includes no other real number. 10. In particular, zero is separated from any positive real by the infinitesimals, and from any negative real by the infinitesimals with negative signs, −dx . 11. Zero is not an infinitesimal, because zero is not strictly greater than zero. 12. We do not add infinity to the hyper-real line. 13. The infinitesimals, the infinitesimals with negative signs, the infinite hyper-reals, and the infinite hyper-reals with negative signs are semi-groups with respect to addition. Neither set includes zero.

14. The hyper-real line is embedded in \∞ , and is not homeomorphic to the real line. There is no bi-continuous one-one mapping from the hyper-real onto the real line.

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15. In particular, there are no points on the real line that can be assigned uniquely to the infinitesimal hyper-reals, or to the infinite hyper-reals, or to the non-constant hyper- reals. 16. No neighbourhood of a hyper-real is homeomorphic to

an \n ball. Therefore, the hyper-real line is not a manifold. 17. The hyper-real line is totally ordered like a line, but it is not spanned by one element, and it is not one-dimensional.

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Integral of a Hyper-real Function

In [Dan3], we defined the integral of a Hyper-real Function. Let f()x be a hyper-real function on the interval [,ab].

The interval may not be bounded. f()x may take infinite hyper-real values, and need not be bounded. At each ax≤≤b, there is a rectangle with base [,xx−+dx dx ], height f()x , and area 22 f()xdx.

We form the Integration Sum of all the areas for the x ’s that start at x= a, and end at xb= , ∑ f ()xdx. xab∈[,]

If for any infinitesimal dx , the Integration Sum has the same hyper-real value, then f ()x is integrable over the interval [,ab].

Then, we call the Integration Sum the integral of f ()x from xa= , to x= b, and denote it by

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xb= ∫ f ()xdx. xa= If the hyper-real is infinite, then it is the integral over [,ab],

If the hyper-real is finite,

xb= ∫ fxdx( )= real part of the hyper-real ., xa=

2.1 The countability of the Integration Sum

In [Dan1], we established the equality of all positive infinities: We proved that the number of the Natural Numbers,

Card` , equals the number of Real Numbers, Card\ = 2Card` , and we have

22Card` Card` Card``=====( Card ) .... 2 2 ... ≡∞. In particular, we demonstrated that the real numbers may be well-ordered. Consequently, there are countably many real numbers in the interval [,ab], and the Integration Sum has countably many terms.

While we do not sequence the real numbers in the interval, the summation takes place over countably many f ()xdx.

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The Lower Integral is the Integration Sum where f ()x is replaced by its lowest value on each interval [,xx−+dx dx ] 22 ⎛⎞ ⎜ ⎟ 2.2 ∑ ⎜ inff (tdx )⎟ ⎜ xtx−≤≤+dx dx ⎟ xab∈[,]⎝⎠22

The Upper Integral is the Integration Sum where f ()x is replaced by its largest value on each interval [,xx−+dx dx ] 22 ⎛⎞ ⎜ ⎟ ⎜ supf (tdx )⎟ 2.3 ∑ ⎜ ⎟ ⎜ xtx−≤≤+dx dx ⎟ xab∈[,]⎝⎠22

If the integral is a finite hyper-real, we have

2.4 A hyper-real function has a finite integral if and only if its upper integral and its lower integral are finite, and differ by an infinitesimal.

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Delta Function

In [Dan5], we have defined the Delta Function, and established its properties 1. The Delta Function is a hyper-real function defined from the

⎪⎧⎫1 ⎪ hyper-real line into the set of two hyper-reals ⎨⎪0, ⎬⎪. The ⎩⎭⎪⎪⎪dx ⎪ hyper-real 0 is the sequence 0, 0, 0, ... . The infinite hyper-

1 real depends on our choice of dx . dx 2. We will usually choose the family of infinitesimals that is

1 1 1 spanned by the sequences , , ,… It is a n n2 n3 semigroup with respect to vector addition, and includes all the scalar multiples of the generating sequences that are non-zero. That is, the family includes infinitesimals with

1 negative sign. Therefore, will mean the sequence n . dx Alternatively, we may choose the family spanned by the

1 1 1 1 sequences , , ,… Then, will mean the 2n 3n 4n dx

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sequence 2n . Once we determined the basic infinitesimal

dx , we will use it in the Infinite Riemann Sum that defines an Integral in Infinitesimal Calculus. 3. The Delta Function is strictly smaller than ∞ 1 4. We define, δ()xx≡ χ⎡⎤−dx, dx (), dx ⎣⎢⎥22⎦

⎧⎪1,x ∈−⎡ dx , dx ⎤ ⎪ ⎣⎢ 22⎦⎥ where χ⎡⎤−dx, dx ()x = ⎨ . ⎣⎦⎢⎥22 ⎪ 0, otherwise ⎩⎪ 5. Hence, for x < 0 , δ()x = 0

dx 1 at x =− , δ()x jumps from 0 to , 2 dx 1 for x ∈−⎡⎤dx, dx , δ()x = . ⎣⎦⎢⎥22 dx 1 at x = 0 , δ(0) = dx dx 1 at x = , δ()x drops from to 0. 2 dx for x > 0 , δ()x = 0 .

xxδ()= 0

1 6. If dx = , δ()xxx= χχχ[,]−−−11 (),2 [,]11 (),3 [,]11 ()...x n 22 44 66

12 3 7. If dx = 2 , δ()x = , , ,... n 2cosh22xxx 2cosh 2 2cosh 2 3

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8. If dx = 1 , δχχχ()xe= −−xxx,2 e23 , 3 e − ,... n [0,∞∞∞ ) [0, ) [0, )

x =∞ 9. ∫ δ()xdx= 1. x =−∞

k=∞ 1 10. δξ()xe−= ik() x−ξ dk 2π ∫ k=−∞

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δ()x and the Hilbert Transform

1 4.1 H{}δ()⋅=− πx Proof: For any infinitesimal dξ , the Integration Sum for the function 111 δξ()= ()ξ χ[,]−ddξξ ξξξ−−xxd22 has only the unique hyper-real term

11 1 ()ξξd = ()ξ. χχ[,]−−ddξξ [,]ddξξ ξξ−−xd 22 ξx 22

Therefore, the Hilbert Transform

ξ=∞ 11 H{}δδ()⋅=pv . . ()ξdξ πξ∫ − x ξ=−∞

11 exists, and equals the constant hyper-real part of ddξξ()ξ , χ[,]− πξ− x 22

11 1 ()ξ =− . χ[,]−ddξξ πξ− xx22 ξ=0 π That is

1 H{}δ()⋅=− ., πx

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Consequently,

1 4.2 δ()x = the inverse Hilbert Transform of the function − πx

t=∞ 11 = pv.. dt. 2 ∫ tt()− x π t=−∞

t=∞ 111 Proof: δ()xpv=− .. − dt ππ∫ tt− x t=−∞ t=∞ 11 = pv.. dt., 2 ∫ tt()− x π t=−∞

t=∞ 11 1 4.3 pv.. dt == an infinite hyper-real π2 ∫ tt()− x dx t=−∞ x =0

t=∞ 11 pv.. dt = 0 π2 ∫ tt()− x t=−∞ x ≠0 1 Proof: δ(0) = . dx δ()x = 0., x ≠0

τ=∞ 11 4.4 δξ()−=xpv .. dτ. 2 ∫ ()()τξτ−−x π τ=−∞

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τξο=− τ=−x ο τ=∞ 11 1 1 =++ddττdτ 2 ∫∫∫()()τξτ−−xx ()()τξτ−− ()()τξτ−−x π ττ=−∞ =ξ +οτ=x +ο for any infinitesimal ο .

t=∞ 11 Proof: δξ()−=xpv .. dt 2 ∫ tt([−−ξ x ]) π t=−∞ The change of variable, τ =+tx, gives

τ=∞ 11 = pv.. dτ 2 ∫ ()()ττξ−−x π τ=−∞

τ=∞ 1111⎪⎪⎧⎫ =−pv.. ⎨⎬⎪⎪dτ ., 2 ∫ ξτξτ−−xx⎪⎪ − π τ=−∞ ⎩⎭⎪⎪

τ=∞ 1111⎪⎪⎧⎫ 4.5 δξ()−=xpv .. ⎨⎬⎪⎪− dτ 2 ∫ ξτξτ−−xx⎪⎪ − π τ=−∞ ⎩⎭⎪⎪

Proof: Apply 4.4.,

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Hilbert Integral Theorem

The Fundamental Theorem of the Hilbert Transform Theory is the Hilbert Integral Theorem. It guarantees that the Hilbert Transform and its Inverse are well defined operations, that when inverted, yield the original function. The use of the Delta Function, and Infinitesimal Calculus Integration, are necessary to establish the Hilbert Integral Theorem. Consequently, it cannot be established in the Calculus of Limits which is inadequate for dealing with singularities, and it does not hold in the Calculus of Limits under any conditions.

5.1 Hilbert Integral Theorem

If ux() is hyper-real function,

Then, the Hilbert Integral Theorem holds.

τξ=∞ ⎛⎞=∞ 111()⎜ u ξ ⎟ u( x )=− pv .. ⎜ pv.. dξτ⎟ d πτπξτ∫∫−−x ⎜ ⎟ τξ=−∞ ⎜⎝⎠=−∞ ⎟

Proof:

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In Infinitesimal Calculus, the Integration Sum

ξ=∞ ∫ ux()(ξδξ− )d ξ ξ=−∞ yields ux(). That is, ξ=∞ ux()=−∫ u ()(ξδξ xd ) ξ. ξ=−∞

By 4.4, δξ(− x) equals the Integration Sum

τ=∞ 11 δξ()−=xpv .. dτ, 2 ∫ ()()τξτ−−x π τ=−∞ where the principal value means that τ has to skip both ξ , and x . Substituting in the Integration Sum for ux(),

ξτ=∞ ⎛⎞=∞ 11⎜ 1⎟ ux()= u ()ξτ⎜ pv .. d⎟ dξ ππτξτ∫∫⎜ ()()−−x ⎟ ξτ=−∞ ⎝⎠⎜ =−∞ ⎟

The terms in this Integration Sum are zero whenever ξ ≠ x . The only nonzero term appears when ξ = x . Changing the Summation order, ξ needs to skip τ ,and the principal value is necessary in the ξ integration as well. Therefore,

τξ=∞ ⎛⎞=∞ 111()⎜ u ξ ⎟ u() x= pv .. ⎜ pv.. dξτ⎟ d πτπτξ∫∫−−x ⎜ ⎟ τξ=−∞ ⎝⎠⎜ =−∞ ⎟

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τξ=∞ ⎛⎞=∞ 111()⎜ u ξ ⎟ ux( )=− pv .. ⎜ pv.. dξτ⎟ d ., πτπξτ∫∫−−x ⎜ ⎟ τξ=−∞ ⎝⎠⎜ =−∞ ⎟

Then, the Hilbert transform of u()ξ ,

ξ=∞ 1(u ξ) pv.. dξ , πξ∫ − τ ξ=−∞ converges to a hyper-real function H()τ , some of its values may be infinite hyper-reals, like the Delta Function. And the Inverse Hilbert Transform of H()τ

τ=∞ 1(H τ) − pv.. dτ πτ∫ − x τ=−∞ converges to the hyper-real function ux().

5.2 If ux() is hyper-real function,

Then,

ξ=∞ 1(u ξ) the hyper-real integral pv.. dξ converges to H()τ πξ∫ − τ ξ=−∞

τ=∞ 1(H τ) the hyper-real integral− pv.. dτ converges to ux() πτ∫ − x τ=−∞

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δ+()t , δ−()t , and the Hilbert transform

6.1 Definition of δ+()x , and δ−()x

ω=0 1 δω()xe≡ itω d − 2π ∫ ω=−∞

ω=∞ 1 δω()xe≡ itω d + 2π ∫ ω=0

6.2 δδ()xx=+−+ () δ ()x

Proof: 3.10.,

6.3 For ω > 0 , H{}eie−−ixωω=− ix

That is, for ω > 0 , H delays the phase of e−ixω by − π . 2 Proof: ξ=∞ 1 e−iωξ H{}epv−ixω = .. dξ πξ∫ − x ξ=−∞

In y < 0, consider the contour γ

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By Cauchy Theorem, e−izω dz = 0. ∫? zx− γ

On the semi-circle that bypasses x ,

ee−−izωω⎪⎪⎧⎫iz dz= iπRes⎨⎬⎪⎪ ∫> zx−−⎪⎪ zx ⎩⎭⎪⎪zx=

= ieπ lim −izω zx→

= ieπ −ixω . By Jordan’s Lemma, on the semi-circle in the lower half plane

e−izω dz → 0 , as z →∞. ∫ zx− Therefore, ξ=∞ e−iωξ princial value of dieξπ+=−ixω 0 . ∫ ξ − x ξ=−∞ That is,

ξ=∞ 1 e−iωξ pv.. dξ =− ie−ixω ., πξ∫ − x ξ=−∞

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6.4 H{}δδ−−()xi=− ()x

That is, H rotates δ ()x by − π − 2 Proof:

⎧⎫ω=0 ⎪⎪1 HH{}δω()xe= ⎨⎬⎪⎪itω d − ⎪⎪2π ∫ ⎩⎭⎪⎪ω=−∞

⎧⎫ω=∞ ⎪⎪1 = H ⎨⎬⎪⎪ed−itω ω ⎪⎪2π ∫ ⎩⎭⎪⎪ω=0

ω=∞ 1 = H{}ed−itω ω 2π ∫ ω=0

ω=∞ 1 =−ie−itω dω 2π ∫ ω=0

ω=0 1 =−ieitω dω 2π ∫ ω=−∞

=−ixδ−().,

6.5 For ω > 0 , H{}eieixωω= ix

That is, for ω > 0 , H advances the phase of eixω by π . 2 Proof:

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ξ=∞ 1 eiωξ H{}epvixω = .. dξ πξ∫ − x ξ=−∞

In y > 0, consider the contour γ

By Cauchy Theorem, eizω dz = 0. ∫> zx− γ

On the semi-circle that bypasses x ,

eeizωω⎪⎪⎧⎫iz dz=− iπRes⎨⎬⎪⎪ ∫? zx−−⎪⎪zx ⎩⎭⎪⎪zx=

=−ieπ lim izω zx→

=−ieπ ixω . By Jordan’s Lemma, on the semi-circle in the upper half plane

eizω dz → 0 , as z →∞. ∫ zx− Therefore,

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ξ=∞ eiωξ princial value of dieξπ−=ixω 0. ∫ ξ − x ξ=−∞ That is,

ξ=∞ 1 eiωξ pv.. dξ = ieixω ., πξ∫ − x ξ=−∞

6.6 H{}δδ++()xi= ()x

That is, H rotates δ ()x by π + 2 Proof:

⎧⎫ω=∞ ⎪⎪1 HH{}δω()xe= ⎨⎬⎪⎪itω d + ⎪⎪2π ∫ ⎩⎭⎪⎪ω=0

ω=∞ 1 = H{}editω ω 2π ∫ ω=0

ω=∞ 1 = ieitω dω 2π ∫ ω=0

= ixδ+().,

6.7 H{}(δδδ()xix=−+− () () x)

Proof: 6.4, and 6.6.,

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1 6.8 δδ()xxi−= () +− πx

Proof: By 6.7,

δδ+−()xxix−=− ()H{} δ ()

ξ=∞ 1(δξ) =−ipv. dξ πξ∫ − x ξ=−∞

11⎛⎞⎟ =−i ⎜ ⎟ π ⎝⎠⎜−x ⎟

1 = i ., πx

11 6.9 δδ()xxi=+ () + 2 πx

11 δδ()xxi=− () − 2 πx Proof: 6.2, and 6.8.,

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References

[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities, and the Continuum Hypothesis” in Gauge Institute Journal Vol.6 No 2, May 2010; [Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No 4, November 2010; [Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal Vol.7 No 1, February 2011; [Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis Origin, the Factorization Error, and the Count of the Primes”, in Gauge

Institute Journal of Math and Physics, November 2009.

[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal Vol.7 No 2, May 2011; [Dirac] Dirac, P. A. M. The Principles of Quantum Mechanics, Second Edition, Oxford Univ press, 1935. [Hen] Henle, James M., and Kleinberg Eugene M., Infinitesimal Calculus, MIT Press 1979. [Hosk] Hoskins, R. F., Standard and Nonstandard Analysis, Ellis Horwood, 1990. [Keis] Keisler, H. Jerome, Elementary calculus, An Infinitesimal Approach, Second Edition, Prindle, Weber, and Schmidt, 1986, pp. 905-912 Kyrala, A., Applied Functions of a Complex Variable, Wiley, 1972, p.243. [Laug] Laugwitz, Detlef, “Curt Schmieden’s approach to infinitesimals-an eye- opener to the historiography of analysis” Technische Universitat Darmstadt, Preprint Nr. 2053, August 1999

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[Mikus] Mikusinski, J. and Sikorski, R., “The elementary theory of distributions”, Rosprawy Matematyczne XII, Warszawa 1957. [Rand] Randolph, John, “Basic Real and Abstract Analysis”, Academic Press, 1968. [Riemann] Riemann, Bernhard, “On the Representation of a Function by a Trigonometric Series”. (1) In “Collected Papers, Bernhard Riemann”, translated from the 1892 edition by Roger Baker, Charles Christenson, and Henry Orde, Paper XII, Part 5, Conditions for the existence of a definite integral, pages 231-232, Part 6, Special Cases, pages 232-234. Kendrick press, 2004 (2) In “God Created the Integers” Edited by Stephen Hawking, Part 5, and Part 6, pages 836-840, Running Press, 2005. Saff, E.B. & Snider A.D. “Fundamentals of Complex Analysis for Mathematics Science, and Engineering” 2nd Edition, Prentice-Hall, 1993. p. 415. [Schwartz] Schwartz, Laurent, Mathematics for the Physical Sciences, Addison-Wesley, 1966. [Spiegel] Spiegel, Murray, Laplace Transforms Schaum, 1965. [Temp] Temple, George, 100 Years of Mathematics, Springer-Verlag, 1981. pp. 19-24.