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 +