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1. Definition Integral Equation: An integral equation is an equation is which an unknown appears under one or more integral sign. 2. Linear and Non-Linear Integral equation: An integral equation is called linear if only linear operation are performed in it upon the unknown function. An integral equation which is not linear is known as a non-linear integral equation. Most general type of integral equation is of the form

( ) ( ) ( ) ( ) ( ) . In this case upper limit may be variable or constant. And = + ∫ , lower limit is always constant. 3. If upper limit is constant then integral equation is known as Fredholm integral equation. If upper limit is variable ( ) then integral equation is known as Volterra integral equation. 4. In above expression if () ≠ then integral equation is third kind. And if () = 0 then integral equation is first kind. Again when () = 1 then integral equation is second kind. If () = 0 then integral equation is homogeneous. 5. Singular Integral equation: When one or both limits of integration become infinite or when the kernel become infinite at one or more points within the range of integration, the integral equation is known as singular integral equation. 6. Symmetric kernel: A kernel (, ) is symmetric (hermitian) if (, ) = (, ) in case of real values = . 7. Seprable of Degenerate kernel: (, ) = ∑ () ℎ() note that () can be regarded as linearly independent. 8. Integral equation of the Convolution type: Kernel must be of type (, ) = ( − )

9. Iterated Kernels: Iterated kernels (, ), = 1,2,3, … are defined as: (, ) = (, ) and ( ) ( ) ( ) , = ∫ , , = 2,3, …. 10. Resolvent Kernel or Reciprocal Kernel: It written in place of (, ) to find solution in form (, ; ) Γ(x, t; λ) 11. Eigen Values and Eigen functions: The values of parameter λ for which integral equation has a non-zero solution. λ = 0 is not an eigen value. If () is an eigen function then () is also eigen function on same eigen value. 12. Note that a homogeneous fredholm integral equation of the second kind may, generally, have no eigen value and eigenfunction, or it may not have nay real eigen value or eigen functions. () 13. Important integral: ∫ () = ∫ () Note that LHS is multiple integral of order ()! n and RHS is ordinary integral of order one. 14. Inner product: ( ) ( ) ( ) , = ∫ / ‖ ‖ | ( )| 15. () = ∫ 16. |(, )| ≤ ‖‖‖‖ (Schwarz inequality) ‖ + ‖ ≤ ‖‖ + ‖‖ (Minkowski inequality)

:Conversion of Ordinary into integral equation:

1. Initial value problem: When an ordinary differential equation is to be solved under conditions involving dependent variable and its derivative at the same vale of the independent variable.

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2. Boundary value problem: When an ordinary differentiable equation is to be solved under conditions involving dependent variable and its derivative at two different values of independent variable. Then the problem consideration is said to be boundary value problem.

:Homogeneous Fredholm integral of the second kind with separable or degenerate kernels:

1. Characteristic values: The number = 0 is not eigenvalue since for = 0 yields () = 0 which is zero solution. 2. If the kernel (, ) is continuous in the rectangle ∶ ≤ ≤ , ≤ ≤ , and the number are finite, then to every eigen value there exist a finite number of linearly independent eigenfunctions. The number of such functions is known as index of eigen value. Different eigen value has different indices. 3. A homogeneous Fredholm integral equation may, generally, have no eigen values and eigen functions or it may not have any real eigen value and eigen function. 4. Solution of homogeneous fredholm integral equation of the second kind with separable kernel: ( ) ( ) ( ) ( ) ∑ ( ) ( ) Given = ∫ , and , = () and = ∫ () then 1 − − − ⎡ ⋯ ⎤ − 1 − eigen value is given by () = ⎢ ⎥ = 0 and = ⎢ ⋮ ⋱ ⋮ ⎥ − − ⎣ ⋯ 1 − ⎦ ( ) ( ) and solution is ( ) ∑ ( ) ∫ =

:Method of Successive Approximations:

1. Iterated Kernels or Functions: Let fredholm integral equation of the second kind () = () + ( ) ( ) then Iterated kernels ( ) are defined as follows ( ) ∫ , , , = 1,2,3, … , = ( ) ( ) ( ) ( ) , and , = ∫ , ( , ) , ∫ , ( , ) = 2,3, ….. this can be done for Volterra integral equation of the second kind by putting = . ( ) ( ) ( ) ( ) 2. ℎ interated kernel , satisfies the relation , = ∫ , , where is any positive integer less than . ( ) ( ) ( ) ( ) 3. Let = + ∫ , be given Fredholm integral equation of the second kind. Suppose that (a) Kernel (, ) ≢ 0, is real and continuous in the rectangle R, for which ≤ ≤ , ≤ ≤ also let |(, )| ≤ , (b) () ≢ 0, is real and continuous in the interval I, for which ≤ ≤ . Also let |()| ≤ (c) is constant such that || ≤ ()

Then equation has a unique solution in I and this solution is given by the absolutely and uniformly ( ) ( ) ( ) ( ) ( ) ( ) ( ) convergent series = + ∫ , + ∫ , ∫ , + ⋯

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4. Let ( ) ( ) ( ) ( ) be given Volterra integral equation of the second kind. = + ∫ , Suppose that (d) Kernel (, ) ≢ 0, is real and continuous in the rectangle R, for which ≤ ≤ , ≤ ≤ also let |(, )| ≤ , (e) () ≢ 0, is real and continuous in the interval I, for which ≤ ≤ . Also let |()| ≤ (f) is constant such that || ≤ ()

Then equation has a unique solution in I and this solution is given by the absolutely and uniformly convergent series ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + ∫ , + ∫ , ∫ , + ⋯ 5. Solution of Fredholm integral equation of the second kind by successive approximations, Iterative methods: Consider Fredholm integral equation of the second kind ( ) ( ) ( ) ( ) then ( ) ∑ and solution is ( ) = + ∫ , , ; = (, ) = ( ) ( ) ( ) + ∫ , ; 6. Let (, ; ) be the resolvent kernel of a Fredholm integral equation ( ) ( ) ( ) ( ) = + ∫ , then the resolvent kernel satisfies the integral equation ( ) ( ) ( ) ( ) . , ; = , + ∫ , , ; 7. The series of resolvent kernel (, ; ) , (, ; ) = ∑ (, ) is absolutely and uniformly convergent for all values of and in the circle || < . (,;) 8. The resolvent kernel satisfies the integro-differential equation = ∫ (, ; ) (, ; ) 9. (, ), = 1,2,3, … are iterated kernels and – (, ) = ∑ (, ), then (, ) and (, ) are known as reciprocal if they are both real and continuous in R as satisfy (, ) + (, ) = ( ) ( ) ( ) ( ) ( ) ( ) ∫ , , and , + , = ∫ , , 10. Volterra’s solution of Fredholm integral equation of the second kind given () = () + ( ) ( ) if (1) ( ) is real and continuous in ( ) ∫ , , ≤ ≤ ≤ ≤ , ≠ 0 (2) () is real and continuous in I and () ≠ 0 (3) If a function (, ) reciprocal to (, ) exists, then the given integral equation has unique solution given by ( ) ( ) ( ) ( ) = − ∫ , 11. Solution of Volterra integral equation of the second kind by successive Approximations: Volterra integral equation of the second kind ( ) ( ) ( ) ( ) , as a zero order = + ∫ , approximation to the required solution (), take () = () and find further approximation by relation ( ) ( ) ( ) ( ) = + ∫ , 12. Important Result: (, ; ) be the resolvent kernel of a Volterra integral equation. () = () + ( ) ( ) then resolvent kernel satisfies the integral equation ( ) ( ) ∫ , , ; = , + ( ) ( ) ∫ , , ; ( ) ( ) ( ) ( ) 13. Solution of Voterra integral equation of second kind : = + ∫ , when its ( ) kernel (, ) is of some particular form. (1) (, ) = () + ()( − ) + () + !

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( ) ( ) ⋯ + () then (, ; ) = ,; where (, ; ) is a solution of the differential ()! equation − () + () + ⋯ + () = 0 satisfying the condition = = = ⋯ = = 0 ℎ = and = 1 ℎ = then solution is given by () = () () + ∫ (, ; )(). (2) (, ) = () + ()( − ) + () + ⋯ + ! ( ) ( ) () then (, ; ) = ,; where ℎ(, ; ) is a solution of the differential ()! equation − () + () + ⋯ + ()ℎ = 0 satisfying the condition ℎ = = = ⋯ = = 0 ℎ = and = 1 ℎ = then solution is given by () = ( ) ( ) ( ) + ∫ , ; . 14. Solution of Voterra integral equation of the second kind by reducing to differential equation (by earlier method given above). 15. Voterra integral equation of the first kind can be converted to a Voterra integral equation of the ( ) ( ) ( ) () second kind. As given ∫ , = () then second kind is given as = − (,) (,) ( ) ∫ here p is depend on condition (, ) ≠ 0 (,)

:Classical :

1. Fredholm’s First Fundamenta theorem: The non-homogeneous Fredholdm integral equation of the second kind ( ) ( ) ( ) ( ) where the functions ( ) and ( ) are = + ∫ , ( ) ( ) ( ) ( ) integrable, has a unique solution = + ∫ , ; where the resolvent kernel ( ) (, ; ) is given by (, ; ) = ,; with D() ≠ 0, is a meromorphic function of the complex () variable , being the ration of two entire functions definied by the series (, ; ) = (, ) + , … , () ∑ ∫ ∫ … ∫ … and ! , … , , … , () () = 1 + ∑ ∫ ∫ … ∫ … both which converge for all values of . ! , … , , … , (, ) ⋯ (, ) = ⋮ ⋱ ⋮ which is known as Fredholm determinant. , … , (, ) ⋯ (, ) ( ) ( ) ( ) 2. The solution of the Fredholm homogeneous equation = ∫ , is identically zero. (,;) 3. () = () + ∫ (, )() resolvent kernel is (, ; ) = where (, ; ) = () ( ) (, ) + ∑ (, ) and () = 1 + ∑ () where ! ! (, ) (, ) … (, ) (, ) ( , ) … ( , ) (, ) = ∫ … ∫ … and ⋮ ⋮ ⋮ (, ) (, ) (, )

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(, ) ⋯ (,) ( ) = ∫ … ∫ ⋮ ⋱ ⋮ … the function , ; is called the (,) ⋯ (, ) Fredholm minor and () is called the Fredholm determinant. 4. method to calculate ( ) and . Take ( ) ℎ , = 1, = ∫ , , ≥ ( ) ( ) ( ) ( ) ( ) 1, , = , , , = . , − ∫ (, ) , , ≥ 1 . after getting ( ) the required solution is given by ( ) ( ) ( ) ( ) , ; = + ∫ , ; 5. Note that: above calculation is done in following sequence— , (, ), , (, ), , (, ) . 6. Fredholm’s Second Fundamenta theorem: if is a zero of multiplicity of the function (), ( ) ( ) ( ) then the homogeneous integral equation = ∫ , possesses at least one, and the , … , , , … most , linearly independent solutions () = , = 1,2, … , 1 ≤ , … , , , … ≤ not identically zero. Any other solution of this equation is a linear combination of these solutions. Here, we have to remember the following definition of the Fredholm minor , … … , , … , ,… … , … … () = + ∑ ∫ ∫ … ∫ … , where , … … , … … ! , … … , , … , {} and {}, = 1,2, … , are two sequence of arbitrary variables. ( ) ( ) ( ) ( ) 7. For an inhomogeneous equation = + ∫ , to possess a solution in the case () = 0, it is necessary and sufficient that the given function () be orthogonal to all the eigenfunctions (), = 1,2, . . , of the transposed homogeneous equation corresponding to the eigenvalue . The general solution has the form ,,…,,,… () = () + ∫ ,,…,,,… () + ∑ (), where () are given by ,…,,,… ,…,,,… ,…,,,… () = ,……………………… = 1,2, . . ,…,,,… ,…………………………

Integral Equation with Symmetric Kernels

1. Symmetric kernels: A kernel is called symmetric if it s coincides with its own complex conjugate. Such a kernel is characterized by the identity (, ) = (, ) 2. If the kernel is real then (, ) = (, ). 3. An integral equation with a symmetric kernel is called symmetric equation. 4. An eigen value is simple if there is only one corresponding eigenfunction, otherwise the eigenvalues are degenerate. 5. The spectrum of the kernel (, ) is the set of all its eigen values. Thus spectrum of a symmetric kernel is never emplty.

6. Regularity conditions: The regularity conditions on kernel (, ) are identical. It is an function if the following three conditions are satisfied: (1) | | ∫ ∫ (, ) < ∞, ∀ ∈ [ ] | | [ ] | | , , ∀ ∈ [, ] (2) ∫ (, ) < ∞ ∀ ∈ , (3) ∫ (, ) < ∞ ∀ ∈ [, ]

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7. Inner product: The inner product of two complex function and of real variable , [ ], is denoted by ( ) and defined as ( ) ( ) ∀ ∈ , , , = ∫ (). 8. Two function are called orthogonal if their inner product is zero, that is, and are orthogonal if (, ) = 0. 9. Norm of a function () is denoted by ‖() ‖ and is given by the relation ‖() ‖ = | | ∫ () 10. A function () is said to be normalized if ‖() ‖ = 1

11. Schwarz inequality: If and are function then |(, )| ≤ ‖‖ ‖‖ 12. Minkowski inequality: If and are function then ‖ + ‖ ≤ ‖‖ + ‖‖ 13. Orthonormal system of functions: A finite or infinite set {()} defined on an interval

∀ ∈ [, ] is said to be an orthogonal set if , = 0, ≠ . {()} is orthonormal if 0, ≠ , = 1, =

14. Riesz-Fischer Theorem: If {()} is a given orthonormal system of functions in -space and {} is a given sequence of complex numbers such that the series ∑|| converges, then there exists a unique function () for which are the Fourier coefficients with respect to the orthonormal system {()} and to which the Fourier series converges in the mean, i.e. ‖() − ∑ ()‖ → 0 → ∞. 15. Let {()} be a complete orhtornormal set over ≤ ≤ , and let {()} be a complete orthonormal set over ≤ ≤ . Then the set ()(), ()(), … . . , ()(), … … is a complete two dimensional orthonormal set over the rectangle ≤ ≤ , ≤ ≤ . 16. If a kernel is symmetric then all its iterated kernels are also symmetric. 17. Every symmetric kernel with a norm not equal to zero has at least one eigenvalue. Or If the kernel (, ) is symmetric and not identically equal to zero, then it has at least one eigen value. 18. If the kernel (, ) is real, symmetric and not identically equal to zero, then it has at least one eigenvalue. 19. The eigenvalue of a symmetric kernel are real. 20. The eigenfunctions of a symmetric kernel, corresponding to different eigenvalues are orthogonal. 21. The multiplicity of any nonzero eigenvalue is finite for every symmetric kernel for which | | ∫ ∫ (, ) is finite. 22. The sequence of eigenfunctions of a symmetric kernel can be made orthonormal.

23. The eigenvalues of a symmetric − form a finite or an infinite sequence {} with no finite limit point. 24. The set of eigenvalues of the second iterated kernel coincide with the set of squares of the eigenvalues of the given kernel.

25. If is the smallest eigenvalue of the kernel K, then ∫ ∫ (,)()() = max (,) = max || ‖‖ ‖‖ 26. Let the sequence {()} be all the eigenfunctions of a symmetric − with {} as the |()| sequence of corresponding eigenvalues. Then, the series ∑ converges and its sum is | | bounded by , which is an upper bound of the integral ∫ (, ) .

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27. Let the sequence () be all the eigenfunctions of a symmetric kernel (, ), with {} as the corresponding eigenvalues. Then, the truncated kernel

() ()() (, ) = (, ) − ∑ has the eigenvalues , , … to which () correspond the eigenfunctions (),(), … the kernel (, ) has no other eigenvalues or eigenfunctions.

28. A necessary and sufficient condition for a symmetric − to be separable is that it must possess a finite number of eigenvalues. 29. Hilber-Schmidt Theorem: If can be written in the form ( ) ( ) ( ) , where () = ∫ , ℎ (, ) is a symmetric − and ℎ() is an −function then () can be expanded in an absolutely and uniformly convergent Fourier series with respect to the orthonormal system of eigenfunctions (), (), … . , (), …. of the kernel (, ) :::::: () = ∑ () , where = (, ). The Fourier coefficients of the function () are related to the Fourier coefficient ℎ of the function ℎ() by the relations = and ℎ = (ℎ, ). Where are the eigenvalues of the kernel (, ).

30. Nonnegative definite kernel: A symmetric − (, ) is said to be nonnegative definite if (, ) ≥ 0 ∀ − . 31. Positive definite kernel: A symmetric − (, ) is said to be positive definite if (, ) ≥ 0 and (, ) = 0 ⇒ .

32. Non positive definite kernel: A symmetric − (, ) is said to be Non positive definite if (, ) ≤ 0∀ − . 33. Negative definite kernel: A symmetric − (, ) is said to be Negative definite if (, ) ≤ 0 and (, ) = 0 ⇒ . 34. Indefinite kernel: A symmetric kernel that does not fall into any of the above mentioned four types of kernels, is known as indefinite kernel.

35. A non null, symmetric − (, ) is nonnegative iff all its eigenvalues are positive. 36. It is positive definite iff the above condition is satisfied and, in addition, some (and therefore every) full orthonormal system of eigenfunctions of (, ) is complete.

37. Mercer’s theorem: If a non null, symmetric − is quasi-definite (i.e. when all but a finite number of eigenvalues are of one sign) and continuous, then the series ∑ is ()() convergent and (, ) = ∑ the series being uniformly and absolutely convergent. 38. ( ) ( ) ( ) ( ) , where ( ) is continuous, real and symmetric and is not = + ∫ , , an eigenvalue. Statement of Hilbert-Schmidt Theorem: Let () be generated form a continuous ( ) ( ) ( ) function () by the operator ∫ , , where , is continuous, real and symmetric, so that ( ) ( ) ( ) . Then ( ) can be represented over interval = ∫ , (, ) by a linear combination of the normalized eigenfunctions of homogeneous integral equation ( ) ( ) ( ) , having ( ) as its kernel. = ∫ , , 39. Operator method in the theory of integral equations: Transformation or Fredholm operator K given by ( ) ( ) . As ( ) ( ) ( ) = ∫ , + = + = where is constant, so K is a linear operator.

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40. The operator K is said to be bounded if ‖‖ ≤ ‖‖ for an − (, ), an − and a constant . ‖ ‖ 41. The norm ‖‖ of K is defined as ‖‖ = . . . ‖‖

42. A Transformation K is said to be continuous in and − if, whenever {} is a sequence in the domain of K if it is continuous at every point therein. 43. Note that linear transformation is continuous if it is bounded. 44. The operator K is given by -- ( ) ( ) , is bounded. = ∫ , 45. An operator is said to be completely continuous if it transforms a bounded set into a compact set. 46. Separable kernel (, ) is given by (, ) = ∑ ()() , where () and () are − , is completely continuous. 47. − (, ) is completely continuous. 48. ℎ norms of and of its disjoint are equal.

49. ℎ reciprocal of the modulus of the eigenvalue with smallest modulus for a symmetric − is equal to the maximum value of |(, )| with ‖‖ = 1. 50. symmetric kernel with a norm not equal to zero has at least one eigen value.

:Singular Integral Equations:

1. An integral equation in which the range of integration is infinite, or in which the kernel is ( ) ( ) ( ) discontinuous, is known as a singular integral equation. For example = ∫ sin , () () = ∫ () , () = ∫ are all the singular integral equation of the first kind. √ () 2. Solution of the Abel integral equation: () = ∫ , 0 < < 1, in which () is a () known function while () is to be determined So its solution is given by () = () ∫ . () () 3. General form of the Abel singular integral equation: : () = ∫ , 0 < < 1, ()() where ℎ is strictly monotonically increasing and differentiable in (, ) and ℎ() ≠ 0 So its ()() solution is given by () = ∫ ()() () 4. General form of the Abel singular integral equation: () = ∫ , 0 < < 1, ()() where < < ℎ and ℎ() is strictly monotonically increasing function So its solution is ()() given by () = − ∫ ()() 5. Weakly singular kernel: A Fredholm integral equation with the kernel of the form (, ) = (,) , 0 < < 1 where (, ) is a bounded function. Then (, ) is known as weakly || singular. 6. Cauchy General and Principal values: In case of improper integral ∫ (), where () is unbounded at = , but is bounded in each of the intervals (, − ) and ( + , ) where and are arbitrary small positive numbers. Then the limit ∫ () = lim→ ∫ () + ∫ (), if it exist, is called the general value of → 26

the improper integral. Note if = = then limit is known as principal value of the improper integral. 7. General value of ∫ () is defined by the limit, ∫ () = lim → ∫ (). And → if = = then limit is known as principal value of the improper integral. 8. Holder Continuous: A function () is said to satisfy the Holder condition if there exist

constant and , 0 < ≤ 1, such that, for every pair of points , lying in the range ≤ ≤ , we have |() − ()| < | − | . A function satisfying the Holder condition is known as Holder continuous. In particular, when = 1, then condition is known as Lipschitz condition. 9. Kernel (,) is Holder continuous with respect to both variables if there exist constants , 0 < < 1 , such that |(, ) − (,)| < [| − | + | − | ] , where (, ) and (, ) lie within the range of definition. () 10. Cauchy : The integral equation () = ∫ , where C is a regular curve, is known as a Cauchy type integral. 11. Premelj Formulas: Let () be a holder continuous function of a point on a regular closed contour C and let a point tend, in arbitrary manner, from inside or outside the contour C, to the () point on this contour: then integral () = ∫ , tends to the limit () = () + ∗ () ∗ () ∫ , or () = − () + ∫ .() relates to the values of the Cauchy integral the region bounded by C, while the second boundary value () relates to the value in the outside region. 12. Poincare-Bertrand Transformation Formula: Let () be Holder continuous function and let

∗ ∗ () C be a closed contour. Then ∫ ∫ = () () 13. Solution of Cauchy Type Singular integral equation: (1) when there is closed contour C,,, ∗ () solution of integral equation () = () − ∫ , where & are known complex constants, () is a Holder continuous function, and C is regular closed contour,,,, is ∗ () () = () − ∫ …. (2) When there is unclosed contour (Riemann () ∗ () Hilbert Problem) solution of integral equation () = () − ∫ is () = ∗ () () + − ∫ , where & are beginning ()() () and end points of the contour C and = log . 14. Hilbert Kernel: A kernel of the form (, ) = cot , where and are real variables, is known as the Hilbert Kernel. 15. Hilbert Formula: ∫ ∫ () = −() + ∫ () 16. Solution of Hilbert type singular integral equation: () = () − ∫ y(t) where & are known complex constants, is () = () − ∫ () + ∫ (). () ()

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17. Solution of integral equation () = ∫ y(t) is () = C − ∫ () solution is exists only when ∫ () = 0

: Method: 1. Voterra integral equation of second kind ( ) ( ) ( ) ( ) Taking Laplace = + ∫ − of the both sides, we get {} = {} + {() ∗ ()} = {} + {()}{()} and then take inverse laplace of the both side, we get solution. 2. Integral transform is defined as ( ) ( ) ( ) where K(p,t) is known as kernel of = ∫ , the transform 0 < 0 3. Laplace transformation: (, ) = then () = ∫ () and denoted ≥ 0 as L{F(t)}

4. {() + ()} = {()} + {()} 5. (1){1} = (2) {} = () > 0 > −1 (3) {} = (4) {cos } = (5) {sin } = (6){cosh } = (7) {sinh } = 6. First Shifting theorem: if {()} = (), ℎ { ()} = ( − ) ( − ), > 7. Second shifting theorem: {()} = () () = ℎ {()} = 0, < () 8. Change of scale property: {()} = (), ℎ {()} = 9. {()} = {()} − (0) 10. { ()} = {()} − ∑ (0) 11. of integral: {()} = () ℎ ∫ () = () 12. {()} = () ℎ {()} = (−1) () () 13. {()} = () ℎ = ∫ () 14. Inverse Laplace transform: F(t)= {()} ( − ), > 15. If {()} = F(t) then {e ()} = () = 0, < 16. If {()} = F(t) then {( − )} = () 17. If {()} = F(t) then {()} = 18. If {()} = F(t) then {()} = (−1)() () 19. If {()} = F(t) then ∫ () = 20. If {()} = F(t) then () = ∫ () 21. If { ( )} ( ) and { ( )} ( ) ( ( ) ( )} ( ) = F t = ℎ . = ∫ ( − ) ≡ ∗ 22. Heaviside’s Expansion theorem: If F(p) and G(p) be two polynomials in p s.t () < () ( ) deg () if G(p) = ( − )( − ) … . . ( − ) then = = ∑ () ()

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