Regularity of the transform of Laplace and the transfom of Fourier

DOI:

By definition, the LZ (t)0(x) function is the transform of Laplace

∞

LZ (t)(P)

• I    e^-xtZ(t)dt,x E [0, to), 0

  and

  
  

  ∞

  
  

  L ± Z (t)()(x) = У

  
  

  e ±$^t z (t)dt, x E [0, to), L - = L,

  
  

  ∞

  F ± u^t)()(p) = У

  
  

  e ± "^ u(t)dt,p E ( -to , to),

  
  

  -∞

  
  

  C ⁰ u«)(x) = J u(t)

  
  

  cos xtdt,

  
  

  ∞

  S ⁰u(t)()(x) = j u(t) sinxt dt, x E (-to, to),

  
  

  ∞

  
  

  FMt)G)(p) = У

  
  

  e ±p«^t u(t)dt,p E ( -to , to ) .

  
  

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 = { ^< p < e U \I^p\ < e} for some e > 0, u(0) = 0, and max[|u(p)|, ldu(p)/dpl, |d2u(p)/p2|]|p2+5| ^ 0, |p| ^ to,

5 > 0, 5 = const., Reu(x) = u(x), x E (-to, to) .

The R1 condition takes place for the l 1 (p), L 1 (p), l ₂ (p), L 2 (p) functions, if the l 1 (p), L 1 (p) functions are regular in left part of the complex plane for all Re <  0, the l ₂ (p),L 2 (p) functions are regular in right part of the plane for all Re >  0 , if the l 1 (p), L 1 (p), l ₂ (p), L 2 (p) functions are continuous for all p E ( - г то ,г то ) from the side of definition of the functions, and the l 1 (p), L 1 (p), l ₂ (p), L 2 (p) functions are limited in the area of the definition (with the values on the ( - г то ,г то ) boundary):

sup / 2 (p) <  c o , sup L 2 (p) <  c o ; Re >0             Re >0

sup I 1 (p) <  c ₀ , sup L 1 (p) <  c ₀ ,c ₀ = const, C ₀ <  TO . Re <0             Re <0

Proposition 1.

li(p) - Li(p) = L2(p) - l2(p) = 0,p = it,t E (-TO, to), if the R1 condition takes place for the l1(p), L1(p),l2(p), L2(p) functions, and li(p) + l2(p) = Li(p) + L2(p),p = it,t E (-то, to).

Proof. We use / 1 (p) — L 1 (p) = L 2 (p) — / 2 (p),p = it,t E ( —то , то ). From the R1 condition the L 2 (p) — / 2 (p) function is an analytical continuation of the / 1 (p) — L 1 (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 ( — i то ,i то ) (by the R1 condition). The function is regular in the full complex plane and is limited in the plane. We obtain / 1 (p) — L i (p) = L 2 (p) — Z ₂ (p) = const = 0. [ ? ; ? ].

The proposition 1 is proved.

In the lemma 1 and the theorem 1 we consider the / 1 (p), L 1 (p), / ₂ (p), L 2 (p) functions:

Mp) = ^L + F - u ⁽ t ⁾⁽^⁾⁽ p ⁾ , Rep <  0; ^⁽p) = ^LF + n^(t)( ^ ⁾⁽p⁾, Rep> 0,

Li(p) = L+^-M-(t)(^)(p),Rep < 0;L2(p) = LF+u- (t)(^)(p),Rep > 0, where ,by definition, u-(t) = u(t), t E [0, +то), u-(t) = —u(t), t E (—то, 0), and for the u(p) function the Y1 condition takes place.

Lemma 1.   1) For the /1(p), L1(p),/2(p), L2(p) functions the R1 condition takes place, if for the u(p) function the Y1 condition takes place.

• 2)    / 1 (p) + / 2 (p) = L 1 (p) + L 2 (p) = 2nu(p/i) , p E G_e , if for the u(p) function the Y1 condition takes place.

Proof. The / 1 (p), L 1 (p), L 2 (p), / ₂ (p) functions (in the form of the Laplace transform) are limited in the area of its definition and on the complex axis ( — i то ,i то ) (the fact is well-known [3;4; 6; 7], if u(0) = 0, we use the Y1 condition [ ? ; ? ] with help of the formula of integration by parts in the inlying integral for the / 1 (p) = L + F - u(t)0(p), L 1 (p) = = L + F - u - (t)( ^ )(p),/ 2 (p), L 2 (p) functions [8]).

From the same formula of integration by parts we obtain, that the / 1 (p), L 1 (p), L 2 (p), / 2 (p) functions are continuous for all p E ( — i то ,i то ) from the side of definition of the functions, if u(0) = 0.

It is obvious, the / 1 (p), L 1 (p), L 2 (p), / ₂ (p) functions are regular in the area of its definition without the ( — i то ,i то ) 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 o f the lemma 1 i s prov ed.

From / 1 (ip) = / 2 (ip),L 1 (ip) = L 2 (ip),p E [0, + то ), we obtain

2Re /1 (ip) = /1 (ip) + /2 (ip), Li (ip) + L2 (ip) = 2Re Li (ip), p = ip, p E [0, +то). We can use the inverse operator of the transform of Fourier [1], and 2Re/1(ip) = 2nu(p), 2ReL1(ip) = 2nu-(p), p E [0, +то), where u(p) = u-(p), p E E [0, +то). We get

/ i (ip) + / 2 (ip) = L i (ip) + L 2 (ip) = 2nu(p),p E [0, + то ).

It is well-known, that the / 1 (p), / ₂ (p) functions are regular for all p : p E {| Rep | < < Ли | Zmp | < 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 I 1 (p), L 1 (p), I 2 (p), L 2 (p) functions are continuous for all p E ( — г то , г то ) from the side of definition of the functions .

We get L 1 (p) = 2nu(p/i) — L 2 (p), p E G _t as the analytical continuation of the L 1 (p) functions [2]) across the [0, +to) axis, and

^ i (p) + ^(p) = L i (p) + L 2 (p) = 2nu(p/i),p E G a , A < г,

• [2 ] (the equality takes place in the area of regularity of the u(z ) function too, z = p/г).

The second part of the lemma 1 is proved.

Theorem 1. S 0 C 0 u(t)( ^ )(x) + C 0 S 0 u(t)( ^ )(x)) = 0 , x E [0, +to) , if for the u(p) function the Y1 condition takes place, and Reu(t) = u(t),t E [0, to) .

Proof. To use the proposition 1 we mast prove, that 1 1 (гу) + 1 ₂ (гу) = L 1 (iy) + Ь ₂ (гу'),у E E ( — г то ,г то ).

From the lemma 1 we get the equality for у E [0,г то ).

For у E ( — г то , 0) we will prove the remark 1.

Remark 1. The LLu(x)( ^ )(z) function is regular in { z : — A < Re z < A } for some A > 0, if for the u(p) function the Y1 condition takes place with г > A (we obtain, that the LLu(x)0(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 we get from the LLu(x)0(z) = J[u(x)/(z + x)\dx equality after a change of the limits of 0

integration in the LLu(x)0(z) function,Re z > 0 [8] (the derivative of LL(z) is defined in Rez> 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 U _A ^ (p) function, U (p) = U(p — A + + A), U _A ^ (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) = U _A ^ (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) = U _A ^ (p) function is the result of reflection of the U(p) function in relation to the (A, 0) point, and the Y (p) function is the result of reflection of the U(p — 2A) 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) = U _A ^ (p) equality takes place for the (A, 0) center of co-ordinates, z = p — A, Rep > A — a >  0, a E (—to, to) , 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) = U _A ^ (p), where U (2A — — p) = U(p — 2A), if and only if U (z) = U ( — z) ((0,0) is the center of coordinates).

The U _A ^ (p) = U (2A — p) = V (p) equality takes place as the definition of the U _A ^ (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 У (p) function:

У (p) = U ((4A — p) — 2A) = U (2A — p) = V (p).

The second part of proposition we obtain from the У(p) = V (p) equality for the new center of co-ordinates in the (A, 0) point. In the U( — (z — A)) = U _A ^ (p) equality U _A ^ (p) = U( — z) , z = p — A, Re z >  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)) = U _A ^ (p) = U( — z ) with U ( — z) = = LL(z ) we obtain, that the LL(z) is regular in Re z >  — 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 L 1 (iy) + L ₂ (iy) = 2nu(y),y E [0,i TO ).

From the remark 1 the L 1 (p) function is regular in p : p E {— A <  Rep < A > 0 } (we use, that the L 1 (p) function is equals to the sum of the two values of the double transform of Laplace [1; 8]).

We get L 1 (p) + L 2 (p) = 2nu(p/i),p E G _A for some A > 0 [2], and L 1 (iy) + L 2 (iy) = = 2nu(y),y E ( — i ^ , 0).

It is well-known, that the l 1 (p),l ₂ (p) functions from the lemma 1 are regular for all p : p E {| Rep | < A J \ Imp \ < A > 0 } , if the u(p) function is regular in p : p E {| Rep | < <  e U l lmp l <  e } for some e > A > 0 [2-4].

The l i (iy) + k(iy) = L i (iy) + L 2 (iy),y E ( — i w ,i w ), is proved.

We can use the proposition 1, and

• ^l 1 Ы ^{— L} 1 ⁽p) = ^L 2 ⁽p) ^{— l} 2 ⁽p) = °-

• (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

∞

=2

⁰ = ^l 2 ⁽ p)

∞∞

j ^.j

^{— L} 2 ⁽p) = ²

e ^ltx u( — x)dx

je^» dtj

e^{tt 5C} u(x)dx =

-

∞

— i[S 0 C 0 u( — t)( ^ )(x) + C 0 S 0 u( — tOx))],

p = ix, x E [0, + ro ), (with u( — t) = u(t), t E [0, to) ) [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.