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DOI: https://doi.org/10.15688/mpcm.jvolsu.2020.4.2
UDC 517.442 Submitted: 15.07.2020
LBC511 Accepted: 06.11.2020
REGULARITY OF THE TRANSFORM OF LAPLACE AND THE TRANSFOM OF FOURIER
Andrey V. Pavlov
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics-1, MIREA — Russian Technological University a_pavlov@mirea.ru
Prosp. Vernadskogo, 78, 119454 Moscow, Russian Federation
Abstract. Regularity of the transform of Laplace in the opened area of 0 is proved with the help of some methods of the transform of Fourier. The class of the transform of Laplace from the transform of Fourier is considered from some functions without a regularity in null. The functions are regular in the opened area of 0. It is proved that the sine transform of Fourier from the cosine transform of Fourier is equal to the cosine transform from the sine transform of Fourier on the module.
Key words: transform of Fourier, transform of Laplace, regularity of the double transform of Laplace, transposition of the sine and cosine transforms of Fourier.
Introduction
In article the methods of complex analysis are applied to research of the transform of Fourier. The main part of the work is considered in the remark 1 and the proposition 2. As a result we obtain a regularity of the double transform of Laplace in some open area of null [3;4;6;7] (the remark 1); by definition, the f (z) function is regular in the z point, if in the o point exists the df (z)/dz derivative. In the [7] work from the fact is proved, that the sine S transform is equal to the cosine transform of Fourier on the module (in the article the fact >r is not proved fully). The methods of research of the article were first considered in the [3;4] work of author. The proposition 1 in the non-obvious form was first considered in the [4] is article. From the propositions we obtain, that the sine transform of Fourier from the cosine transform of Fourier is equal to the cosine transform from the sine transform of Fourier on @the module (theorem 1).
By definition, the LZ(t)(-)(x) function is the transform of Laplace
oo
— xt I
LZ(t)^)(x) = J e Z(t)dt,x E [0, ж), 0
and
oo
L±Z(t)^)(x) = J e±xtZ(t)dt, x E [0, ж),L- = L 0
o
F±u(t)^)(p) = f e±pitu(t)dt,p E (-ж, ж),
-o
o o
00
C u(t)(-)(x) = J u(t) cosxtdt, S u(t)(-)(x) = J u(t)sinxtdt,x E (—tt, tt), 0 0 oo
F0u(t)(-)(p) = J e±pitu(t)dt,p E (—tt, tt). o
1. Regularity of the transform of Laplace and the transform of Fourier
In the section we use the Y1 and R1 conditions.
The Y1 condition takes place for the u(p) function, if the u(p) function is regular in p : p E Ge = {\Rep\ < £ U |Imp\ < e} for some e > 0, u(0) = 0, and
max[\u(p)\, \du(p)/dp\, \d2u(p)/p2\]\p2+b\ ^ 0, \p\ ^ tt,
6 > 0, 6 = const., Re u(x) = u(x),x E (—tt, tt).
The R1 condition takes place for the ll(p),Ll(p), l2(p),L2(p) functions, if the l\(p), Ll(p) functions are regular in left part of the complex plane for all Re < 0, the l2(p),L2(p) functions are regular in right part of the plane for all Re > 0 , if the li(p),Li(p), l2(p),L2(p) functions are continuous for all p E (—itt, itt) from the side of definition of the functions, and the ll(p),Ll(p), l2(p),L2(p) functions are limited in the area of the definition (with the values on the (—itt, itt) boundary):
sup 12(p) < c0, sup L2(p) < c0;
Re>0 Re>0
sup h(p) < Co, sup Li(p) < Co, Co = const, c0 < tt.
Re<0 Re<0
Proposition 1.
Ii(p) — Li(p) = L2(p) — k(p) = 0,p = it,t E (—tt, tt), if the R1 condition takes place for the li(p),Li(p), l2(p),L2(p) functions, and h(p) + 12(p) = Li(p) + L2(p),p = it,t E (—tt, tt).
Proof. We use ll(p) — Li(p) = L2(p) — l2(p),p = it,t E (—m, <x>). From the R1 condition the L2(p) — l2(p) function is an analytical continuation of the li(p) — Li(p) function from the left to the right part of the plane across the complex axis [2]. The functions are limited in area of its definition and on the complex axis (—im,im) (by the R1 condition). The function is regular in the full complex plane and is limited in the plane. We obtain h(p) — Li(p) = L2(p) — l2(p) = const = 0. [?; ?]. The proposition 1 is proved.
In the lemma 1 and the theorem 1 we consider the ll(p),Ll(p),l2(p),L2(p) functions:
h(p) = L+F-u(t)(^)(p),Rep < 0; k(p) = LF+u(t)(^)(p), Rep > 0,
Li(p) = L+F-U-(t)^)(p),Rep < 0; L2(p) = LF+U-(t)(^)(p), Rep > 0, where ,by definition,
u-(t) = u(t),t E [0, +m),U-(t) = —u(t),t E (—m, 0),
and for the u(p) function the Y1 condition takes place.
Lemma 1. 1) For the ll(p),Ll(p),l2(p),L2(p) functions the R1 condition takes place, if for the u(p) function the Y1 condition takes place.
2) h(p) + l2(p) = Ll(p) + L2(p) = 2nu(p/i), p E Ge, if for the u(p) function the Y1 condition takes place.
Proof. The ll(p), Ll(p), L2(p), l2(p) functions (in the form of the Laplace transform) are limited in the area of its definition and on the complex axis (—im,im) (the fact is well-known [3;4;6;7], if «(0) = 0, we use the Y1 condition [?; ?] with help of the formula of integration by parts in the inlying integral for the ll(p) = L+F-u(t)(^)(p), Ll(p) = = L+F-U-(t)(^)(p),l2(p),L2(p) functions [8]).
From the same formula of integration by parts we obtain, that the ll(p), Ll(p), L2(p), l2(p) functions are continuous for all p E (—i<x>,i<x>) from the side of definition of the functions, if m(0) = 0.
It is obvious, the ll(p), Ll(p), L2(p), l2(p) functions are regular in the area of its definition without the (—i<x>,i<x>) boundary (in the points exists the df (p)/dp derivative, if for the u(p) function the Y1 condition takes place [8]). The first part of the lemma 1 is proved. From h(iy) = l2(iy), Ll(iy) = L2(iy),y E [0, +m), we obtain
2Reh(iy) = h(iy) + k(iy),Ll(iy) + L2(iy) = 2ReLl(iy),p = iy,
y E [0, +m). We can use the inverse operator of the transform of Fourier [1], and 2Rell(iy) = 2nu(y), 2ReLl(iy) = 2nu-(y), y E [0, +m), where u(y) = u-(y), y E E [0, +m). We get
h(iy) + h(iy) = Ll(iy) + L2(iy) = 2nu(y),y E [0, +m).
It is well-known, that the ll(p), l2(p) functions are regular for all p : p E {iRepI < < AU IlmpI < A> 0} , if the Y1 condition for the u(p) function takes place for some e > A > 0 [2-4].
We can use, that from the R1 condition the h(p), Ll(p), l2(p), L2(p) functions are continuous for all E (— tt, tt) from the side of definition of the functions .
We get Ll(p) = 2nu(p/%) — L2(p), p E Ge as the analytical continuation of the Ll(p) functions [2]) across the [0, +tt) axis, and
h(p) + 12(p) = Li(p) + L2(p) = 2nu(p/i),p ega,a< e,
[2] (the equality takes place in the area of regularity of the u( ) function too, = / ). The second part of the lemma 1 is proved.
Theorem 1. S0C0u(t)(-)(x) + C0S0u(t)(-)(x)) = 0, x E [0, +tt), if for the u(p) function the Y1 condition takes place, and Reu(t) = u(t),t E [0, tt).
Proof. To use the proposition 1 we mast prove, that h(iy) + 12(iy) = Ll(iy) + L2(iy),y E E ( — 1tt, itt).
From the lemma 1 we get the equality for y E [0, itt). For E (— tt, 0) we will prove the remark 1.
Remark 1. The LLu(x)(-)(z) function is regular in {z : —A < Rez < A} for some A> 0, if for the u(p) function the Y1 condition takes place with e > A (we obtain, that the LLu(x)(-)(z) function is non multiple-meaning function in {z : —A < Imz < A} [2; 4]).
Proof. The LL(z) function is regular in Rew > 0, if LL(w) = LLu(x)(-)(z). The fact
oo
we get from the LLu(x)(-)(z) = j[u(x)/(z + x)]dx equality after a change of the limits of
o
integration in the LLu(x)(-)(z) function,Rez > 0 [8] (the derivative of LL(z) is defined in R > 0 in the Y1 condition [2]).
The main part of the remark 1 is the proposition 2.
In formulation of proposition 2 the new definitions are used: the reflection of the U(p) function in relation to the (A, 0) point as the ua^(p) function, U(p) = U(p — A + + A), UA^(p) = U(—(p — A) + A) = U(2A — p) ( (0,0) is the center of co-ordinates); the 2A-moving of the U(p) function is the U(p — 2A) function. It is obviously, if we will move the U(p) function to the right on 2A value we obtain the U(p — 2A) function. Proposition 2. 1) U (p — 2A) = UA^(p), \p— 2A\ < A> 0 if and only if U (—p) = U (p), \p\ < A> 0, if the U (p) function is defined in \p\ < A.
2) The V (p) = Y (p), \p — 2A\ < A> 0 equality takes place, where the V (p) = ua^(p) function is the result of reflection of the U( ) function in relation to the ( A, 0) point, and the Y( ) function is the result of reflection of the U( — 2 A) function in relation to the (2A, 0) point (the (0,0) point is the center of co-ordinates, and the U(p) function is defined in \p\ < A). The U(A — z) = ua^(p) equality takes place for the (A, 0) center of co-ordinates, z = p — A, Rep > A — a > 0,a E (—tt, tt), if the U(p) function is defined in Rep > A + a (for all A > 0).
Proof. The first part of proposition we obtain from U(2A — p) = ua^(p), where U(2A — — p) = U(p — 2A), if and only if U(z) = U(—z) ((0,0) is the center of coordinates).
The ua^(p) = U(2A — p) = V(p) equality takes place as the definition of the ua^(p) reflection ((0,0) is the center of co-ordinates). The 2A-moving of the U(p) function is the
U(p — 2A) function; the result of the reflection of the U(p — 2A) function in relation to the (2A, 0) point is the Y(p) function:
Y (p) = U ((4A — p) — 2 A) = U (2A — p) = V (p).
The second part of proposition we obtain from the Y(p) = V(p) equality for the new center of co-ordinates in the (A, 0) point. In the U(—(z — A)) = UA^(p) equality UA^(p) = U(—z) , z = p — A,Rez> 0, if we consider the U(—z), U(—(z — A)) function as the comparison of points of the complex plane to the points on other plane (with z = p — A). The proposition 2 is proved.
With help of the proposition 2 from U(—(z — A)) = UA^(p) = U(—z) with U(—z) = = LL(z) we obtain, that the LL(z) is regular in Rez > —A < 0, if the U(—z) function is regular for all z : Rez > 0 [2] (the U(z) function from the second part of the proposition 2 is defined for all Rep > A + a, a = A). The remark 1 is proved.
From the second part of the lemma 1 we obtain Li(iy) + L2(iy) = 2nu(y),y E [0,zro). From the remark 1 the Li(p) function is regular in p : p E {—A < Rep < A > 0} (we use, that the Li(p) function is equals to the sum of the two values of the double transform of Laplace [1; 8]).
We get Li(p) + L2(p) = 2nu(p/i),p E GA for some A> 0 [2], and Li(iy) + L2(iy) = = 2nu(y),y E (—i<x>, 0).
It is well-known, that the h(p),l2(p) functions from the lemma 1 are regular for all p : p E {iRepI < ^U llmpl < A> 0} , if the u(p) function is regular in p : p E {iRepI < < e U IlmpI < £} for some £ > A> 0 [2-4].
The h(iy) + l2(iy) = Li(iy) + L2(iy),y E (—i<x>, i<x), is proved. We can use the proposition 1, and
h (P) — Ll(j>) = L2(p) — h(p) = 0.
(All other conditions of the proposition 1 are the result of the first part of the lemma 1). The result of the theorem 1 we obtain from
^ 0
0 = k(p) — L2(p) = 2 J e-ptdt J eitxu(x)dx =
0 -TO
TO TO
= 2 J e-ptdtj e-itxu(—x)dx = —i[S0C0u(—t)(^)(x) + C0S0u(—t)(^)(x))], 00
p = ix,x E [0, (with u(—t) = u(t),t E [0, [2; 3; 5-8].
Conclusion
Results of the theorem 1 and proposition 2 probably are of interest for the further study from point of physical applications.
REFERENCES
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3. Pavlov A.V. About the Equality of the Transform of Laplace to the Transform of Fourier. Issues of Analysis, 2016, vol. 23, no. 1, pp. 21-30.
4. Pavlov A.V. The Fourier Transform and New Inversion Formula of the Laplace Transform. Math. notes, 2011, vol. 90, no. 6, pp. 793-796.
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РЕГУЛЯРНОСТЬ ПРЕОБРАЗОВАНИЯ ЛАПЛАСА И ПРЕОБРАЗОВАНИЕ ФУРЬЕ
Андрей Валерианович Павлов
Кандидат физико-математических наук, доцент кафедры высшей математики-1,
МИРЭА — Российский технологический университет
a_pavlov@mirea.ru
просп. Вернадского, 78, 119454 г. Москва, Российская Федерация
Аннотация. Доказана регулярность двойного преобразования Лапласа в окрестности нуля. Рассматривается класс функций с нарушением регулярности в нуле. Преобразование Лапласа от преобразования Фурье от функций из данного класса регулярно в окрестности нуля. Доказана перестановочность синус и косинус преобразований Фурье в условиях регулярности двойного преобразования Лапласа в окрестности нуля.
Ключевые слова: преобразование Фурье, преобразование Лапласа, регулярность двойного преобразования Лапласа, перестановочность синус и косинус преобразований Фурье.
