Handbook of mathematics for engineers and scienteists part 172 potx

T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1165 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 14  f(p) (p + a) 2 + b 2 1 b  x 0 e –a(x–t) sin  b(x – t)  f(t) dt 15 1 p n  f(p), n = 1, 2, 1 (n – 1)!  x 0 (x – t) n–1 f(t) dt 16  f 1 (p)  f 2 (p)  x 0 f 1 (t)f 2 (x – t) dt 17 1 √ p  f  1 p   ∞ 0 cos  2 √ xt  √ πx f(t) dt 18 1 p √ p  f  1 p   ∞ 0 sin  2 √ xt  √ πt f(t) dt 19 1 p 2ν+1  f  1 p   ∞ 0 (x/t) ν J 2ν  2 √ xt  f(t) dt 20 1 p  f  1 p   ∞ 0 J 0  2 √ xt  f(t) dt 21 1 p  f  p + 1 p   x 0 J 0  2 √ xt – t 2  f(t) dt 22 1 p 2ν+1  f  p + a p  ,– 1 2 < ν ≤ 0  x 0  x – t at  ν J 2ν  2 √ axt – at 2  f(t) dt 23  f  √ p   ∞ 0 t 2 √ πx 3 exp  – t 2 4x  f(t) dt 24 1 √ p  f  √ p  1 √ πx  ∞ 0 exp  – t 2 4x  f(t) dt 25  f  p + √ p  1 2 √ π  x 0 t (x – t) 3/2 exp  – t 2 4(x – t)  f(t) dt 26  f   p 2 + a 2  f(x)–a  x 0 f  √ x 2 – t 2  J 1 (at) dt 27  f   p 2 – a 2  f(x)+a  x 0 f  √ x 2 – t 2  I 1 (at) dt 28  f   p 2 + a 2   p 2 + a 2  x 0 J 0  a √ x 2 – t 2  f(t) dt 29  f   p 2 – a 2   p 2 – a 2  x 0 I 0  a √ x 2 – t 2  f(t) dt 30  f   (p + a) 2 – b 2  e –ax f(x)+be –ax  x 0 f  √ x 2 – t 2  I 1 (bt) dt 31  f(ln p)  ∞ 0 x t–1 Γ(t) f(t) dt 32 1 p  f (ln p)  ∞ 0 x t Γ(t + 1) f(t) dt 33  f (p – ia)+  f (p + ia), i 2 =–1 2f(x)cos(ax) 34 i   f(p – ia)–  f(p + ia)  , i 2 =–1 2f(x)sin(ax) 1166 INTEGRAL TRANSFORMS No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 35 d  f(p) dp – xf(x) 36 d n  f(p) dp n (–x) n f(x) 37 p n d m  f(p) dp m , m ≥ n (–1) m d n dx n  x m f(x)  38  ∞ p  f(q) dq 1 x f(x) 39 1 p  p 0  f(q) dq  ∞ x f(t) t dt 40 1 p  ∞ p  f(q) dq  x 0 f(t) t dt T3.2.2. Expressions with Rational Functions No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 1 1 p 1 2 1 p + a e –ax 3 1 p 2 x 4 1 p(p + a) 1 a  1 – e –ax  5 1 (p + a) 2 xe –ax 6 p (p + a) 2 (1 – ax)e –ax 7 1 p 2 – a 2 1 a sinh(ax) 8 p p 2 – a 2 cosh(ax) 9 1 (p + a)(p + b) 1 a – b  e –bx – e –ax  10 p (p + a)(p + b) 1 a – b  ae –ax – be –bx  11 1 p 2 + a 2 1 a sin(ax) 12 p p 2 + a 2 cos(ax) 13 1 (p + b) 2 + a 2 1 a e –bx sin(ax) 14 p (p + b) 2 + a 2 e –bx  cos(ax)– b a sin(ax)  T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1167 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 15 1 p 3 1 2 x 2 16 1 p 2 (p + a) 1 a 2  e –ax + ax – 1  17 1 p(p + a)(p + b) 1 ab(a – b)  a – b + be –ax – ae –bx  18 1 p(p + a) 2 1 a 2  1 – e –ax – axe –ax  19 1 (p + a)(p + b)(p + c) (c – b)e –ax +(a – c)e –bx +(b – a)e –cx (a – b)(b – c)(c – a) 20 p (p + a)(p + b)(p + c) a(b – c)e –ax + b(c – a)e –bx + c(a – b)e –cx (a – b)(b – c)(c – a) 21 p 2 (p + a)(p + b)(p + c) a 2 (c – b)e –ax + b 2 (a – c)e –bx + c 2 (b – a)e –cx (a – b)(b – c)(c – a) 22 1 (p + a)(p + b) 2 1 (a – b) 2  e –ax – e –bx +(a – b)xe –bx  23 p (p + a)(p + b) 2 1 (a – b) 2  –ae –ax +[a + b(b – a)x  e –bx  24 p 2 (p + a)(p + b) 2 1 (a – b) 2  a 2 e –ax + b(b – 2a – b 2 x + abx)e –bx  25 1 (p + a) 3 1 2 x 2 e –ax 26 p (p + a) 3 x  1 – 1 2 ax  e –ax 27 p 2 (p + a) 3  1 – 2ax + 1 2 a 2 x 2  e –ax 28 1 p(p 2 + a 2 ) 1 a 2  1 –cos(ax)  29 1 p  (p + b) 2 + a 2  1 a 2 + b 2  1 – e –bx  cos(ax)+ b a sin(ax)   30 1 (p + a)(p 2 + b 2 ) 1 a 2 + b 2  e –ax + a b sin(bx)–cos(bx)  31 p (p + a)(p 2 + b 2 ) 1 a 2 + b 2  –ae –ax + a cos(bx)+b sin(bx)  32 p 2 (p + a)(p 2 + b 2 ) 1 a 2 + b 2  a 2 e –ax – ab sin(bx)+b 2 cos(bx)  33 1 p 3 + a 3 1 3a 2 e –ax – 1 3a 2 e ax/2  cos(kx)– √ 3 sin(kx)  , k = 1 2 a √ 3 34 p p 3 + a 3 – 1 3a e –ax + 1 3a e ax/2  cos(kx)+ √ 3 sin(kx)  , k = 1 2 a √ 3 1168 INTEGRAL TRANSFORMS No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 35 p 2 p 3 + a 3 1 3 e –ax + 2 3 e ax/2 cos(kx), k = 1 2 a √ 3 36 1  p + a)  (p + b) 2 + c 2 ] e –ax – e –bx cos(cx)+ke –bx sin(cx) (a – b) 2 + c 2 , k = a – b c 37 p  p + a)  (p + b) 2 + c 2 ] –ae –ax + ae –bx cos(cx)+ke –bx sin(cx) (a – b) 2 + c 2 , k = b 2 + c 2 – ab c 38 p 2  p + a)  (p + b) 2 + c 2 ] a 2 e –ax +(b 2 +c 2 –2ab)e –bx cos(cx)+ke –bx sin(cx) (a –b) 2 +c 2 , k =–ac – bc + ab 2 – b 3 c 39 1 p 4 1 6 x 3 40 1 p 3 (p + a) 1 a 3 – 1 a 2 x + 1 2a x 2 – 1 a 3 e –ax 41 1 p 2 (p + a) 2 1 a 2 x  1 + e –ax  + 2 a 3  e –ax – 1  42 1 p 2 (p + a)(p + b) – a + b a 2 b 2 + 1 ab x + 1 a 2 (b – a) e –ax + 1 b 2 (a – b) e –bx 43 1 (p + a) 2 (p + b) 2 1 (a – b) 2  e –ax  x + 2 a – b  + e –bx  x – 2 a – b  44 1 (p + a) 4 1 6 x 3 e –ax 45 p (p + a) 4 1 2 x 2 e –ax – 1 6 ax 3 e –ax 46 1 p 2 (p 2 + a 2 ) 1 a 3  ax –sin(ax)  47 1 p 4 – a 4 1 2a 3  sinh(ax)–sin(ax)  48 p p 4 – a 4 1 2a 2  cosh(ax)–cos(ax)  49 p 2 p 4 – a 4 1 2a  sinh(ax)+sin(ax)  50 p 3 p 4 – a 4 1 2  cosh(ax)+cos(ax)  51 1 p 4 + a 4 1 a 3 √ 2  cosh ξ sin ξ –sinhξ cos ξ  , ξ = ax √ 2 52 p p 4 + a 4 1 a 2 sin  ax √ 2  sinh  ax √ 2  53 p 2 p 4 + a 4 1 a √ 2  cos ξ sinh ξ +sinξ coshξ  , ξ = ax √ 2 T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1169 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 54 1 (p 2 + a 2 ) 2 1 2a 3  sin(ax)–ax cos(ax)  55 p (p 2 + a 2 ) 2 1 2a x sin(ax) 56 p 2 (p 2 + a 2 ) 2 1 2a  sin(ax)+ax cos(ax)  57 p 3 (p 2 + a 2 ) 2 cos(ax)– 1 2 ax sin(ax) 58 1  (p + b) 2 + a 2  2 1 2a 3 e –bx  sin(ax)–ax cos(ax)  59 1 (p 2 – a 2 )(p 2 – b 2 ) 1 a 2 – b 2  1 a sinh(ax)– 1 b sinh(bx)  60 p (p 2 – a 2 )(p 2 – b 2 ) cosh(ax)–cosh(bx) a 2 – b 2 61 p 2 (p 2 – a 2 )(p 2 – b 2 ) a sinh(ax)–b sinh(bx) a 2 – b 2 62 p 3 (p 2 – a 2 )(p 2 – b 2 ) a 2 cosh(ax)–b 2 cosh(bx) a 2 – b 2 63 1 (p 2 + a 2 )(p 2 + b 2 ) 1 b 2 – a 2  1 a sin(ax)– 1 b sin(bx)  64 p (p 2 + a 2 )(p 2 + b 2 ) cos(ax)–cos(bx) b 2 – a 2 65 p 2 (p 2 + a 2 )(p 2 + b 2 ) –a sin(ax)+b sin(bx) b 2 – a 2 66 p 3 (p 2 + a 2 )(p 2 + b 2 ) –a 2 cos(ax)+b 2 cos(bx) b 2 – a 2 67 1 p n , n = 1, 2, 1 (n – 1)! x n–1 68 1 (p + a) n , n = 1, 2, 1 (n – 1)! x n–1 e –ax 69 1 p(p + a) n , n = 1, 2, a –n  1 – e –ax e n (ax)  , e n (z)=1 + z 1! + ···+ z n n! 70 1 p 2n + a 2n , n = 1, 2, – 1 na 2n n  k=1 exp(a k x)  a k cos(b k x)–b k sin(b k x)  , a k = a cos ϕ k , b k = a sinϕ k , ϕ k = π(2k – 1) 2n 71 1 p 2n – a 2n , n = 1, 2, 1 na 2n–1 sinh(ax)+ 1 na 2n n  k=2 exp(a k x) ×  a k cos(b k x)–b k sin(b k x)  , a k = a cos ϕ k , b k = a sinϕ k , ϕ k = π(k – 1) n 1170 INTEGRAL TRANSFORMS No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 72 1 p 2n+1 + a 2n+1 , n = 0, 1, e –ax (2n + 1)a 2n – 2 (2n + 1)a 2n+1 n  k=1 exp(a k x) ×  a k cos(b k x)–b k sin(b k x)  , a k = a cos ϕ k , b k = a sinϕ k , ϕ k = π(2k – 1) 2n + 1 73 1 p 2n+1 – a 2n+1 , n = 0, 1, e ax (2n + 1)a 2n + 2 (2n + 1)a 2n+1 n  k=1 exp(a k x) ×  a k cos(b k x)–b k sin(b k x)  , a k = a cos ϕ k , b k = a sinϕ k , ϕ k = 2πk 2n + 1 74 Q(p) P (p) , P (p)=(p – a 1 ) (p – a n ); Q(p) is a polynomial of degree ≤ n – 1; a i ≠ a j if i ≠ j n  k=1 Q(a k ) P  (a k ) exp  a k x  , (the prime stand for the differentiation) 75 Q(p) P (p) , P (p)=(p – a 1 ) m 1 (p – a n ) m n ; Q(p) is a polynomial of degree < m 1 + m 2 + ···+ m n – 1; a i ≠ a j if i ≠ j n  k=1 m k  l=1 Φ kl (a k ) (m k – l)! (l – 1)! x m k –l exp  a k x  , Φ kl (p)= d l–1 dp l–1  Q(p) P k (p)  , P k (p)= P (p) (p – a k ) m k 76 Q(p)+pR(p) P (p) , P (p)=(p 2 + a 2 1 ) (p 2 + a 2 n ); Q(p)andR(p) are polynomials of degree ≤ 2n – 2; a l ≠ a j , l ≠ j n  k=1 Q(ia k )sin(a k x)+a k R(ia k )cos(a k x) a k P k (ia k ) , P m (p)= P (p) p 2 + a 2 m , i 2 =–1 T3.2.3. Expressions with Square Roots No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 1 1 √ p 1 √ πx 2 √ p – a –  p – b e bx – e ax 2 √ πx 3 3 1 √ p + a 1 √ πx e –ax 4  p + a p – 1 1 2 ae –ax/2  I 1  1 2 ax  + I 0  1 2 ax  5 √ p + a p + b e –ax √ πx +(a – b) 1/2 e –bx erf  (a – b) 1/2 x 1/2  6 1 p √ p 2  x π 7 1 (p + a) √ p + b (b – a) –1/2 e –ax erf  (b – a) 1/2 x 1/2  T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1171 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 8 1 √ p (p – a) 1 √ a e ax erf  √ ax  9 1 p 3/2 (p – a) a –3/2 e ax erf  √ ax  – 2a –1 π –1/2 x 1/2 10 1 √ p + a π –1/2 x –1/2 – ae a 2 x erfc  a √ x  11 a p  √ p + a  1 – e a 2 x erfc  a √ x  12 1 p + a √ p e a 2 x erfc  a √ x  13 1  √ p + √ a  2 1 – 2 √ π (ax) 1/2 +(1 – 2ax)e ax  erf  √ ax  – 1  14 1 p  √ p + √ a  2 1 a +  2x – 1 a  e ax erfc  √ ax  – 2 √ πa √ x 15 1 √ p  √ p + a  2 2π –1/2 x 1/2 – 2axe a 2 x erfc  a √ x  16 1  √ p + a  3 2 √ π (a 2 x + 1) √ x – ax(2a 2 x + 3)e a 2 x erfc  a √ x  17 p –n–1/2 , n = 1, 2, 2 n 1×3× × (2n – 1) √ π x n–1/2 18 (p + a) –n–1/2 2 n 1×3× × (2n – 1) √ π x n–1/2 e –ax 19 1  p 2 + a 2 J 0 (ax) 20 1  p 2 – a 2 I 0 (ax) 21 1  p 2 + ap + b exp  – 1 2 ax  J 0  (b – 1 4 a 2  1/2 x  22   p 2 + a 2 – p  1/2 1 √ 2πx 3 sin(ax) 23 1  p 2 + a 2   p 2 + a 2 + p  1/2 √ 2 √ πx cos(ax) 24 1  p 2 – a 2   p 2 – a 2 + p  1/2 √ 2 √ πx cosh(ax) 25   p 2 + a 2 + p  –n na –n x –1 J n (ax) 26   p 2 – a 2 + p  –n na –n x –1 I n (ax) 27  p 2 + a 2  –n–1/2 (x/a) n J n (ax) 1×3×5× × (2n – 1) 28  p 2 – a 2  –n–1/2 (x/a) n I n (ax) 1×3×5× × (2n – 1) . T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1165 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 14  f(p) (p. + b) 2 + a 2 e –bx  cos(ax)– b a sin(ax)  T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1167 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 15 1 p 3 1 2 x 2 16 1 p 2 (p. a 4 1 a √ 2  cos ξ sinh ξ +sinξ coshξ  , ξ = ax √ 2 T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1169 No. Laplace transform,  f(p) Inverse transform, f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp 54 1 (p 2 +