MA102 : Linear algebra and Integral Transforms Tutorial Sheet-12
Second semester of academic year 2018-19
1. Find the Laplace transform of the function, f(t) = etd2019
dt2019(e−tt2019).
2. Find the Laplace transform of the following functions:
(a) f(t) =eatcoshbt, (b) f(t) = cosatcoshbt,
(c) f(t) =te−4tsin 3t.
3. Find the Laplace transform of the function {sin√
t} and hence deduce that L(g(t)) =π
s 12
e−4s1 , whereg(t) = cos√
√ t t . 4. Using the Laplace transform show that
(a) Z ∞
0
sint
t dt= π 2, (b)
Z ∞
0
te−3tsintdt= 3 50.
5. If L{f(t)}=F(s) and a >0, then show that L−1{F(as+b)}= 1 ae−bta f
t a
.
6. If f(t) is a periodic function with period T then, show thatL[f(t)] = RT
0 e−stf(t)dt 1−e−sT and find the Laplace transform of the following function
f(t) =
(1, 0≤t <2
−1, 2≤t <4, wheref(t+ 4) =f(t).
7. Find the Laplace transform of f(t) where f(t) defined as f(t) =
(t
T, 0< t < T 1, t > T.
8. If L−1[F(s)] = f(t) then, show that L−1[F(as)] = 1 af
t a
. 9. Find inverse Laplace transform of following functions:
(a) log(1 + s12), (b) (s+1)(s12+1),
(c) (s−1)(s5s+32+2s+5).
10. Show that L−1{tan−1 2s2}= 2t(sint)(sinht).
11. Using convolution find (a) L−1[s2(s+1)1 2], (b) L−1[(s+2)12(s−2)].
12. If erf(t) = √2π Rt
0 e−x2dx, then show that L{erf(√
t)}= 1
s√ s+ 1. 13. Evaluate the following. Use Γ(1/2) =√
π, where necessary, a)Γ(3/2) b) Γ(8/3)Γ(3/2) c)Γ(−9/2) d) B(3/4,1/4).
14. Evaluate the following integrals:
(a) R π2
0
√tanx dx,
(b) Z 1
−1
1 +x 1−x
12 dx.
15. Show that Γ(2x) = 22x−1√π Γ(x)Γ(x+12).
16. Show that R∞
0 xme−xn dx= 1 nΓ
m+ 1 n
, (m >−1, n >0).
17. Show that R∞
0 cosx2 dx=R∞
0 sinx2 dx= 12pπ
2. 18. Verify the given integral formulas
(a) Γ(x) =pxR∞
0 e−pt tx−1dt, x >0, p >0, (b) Γ(x) =R∞
−∞ext−etdt, x >0, (c) Γ(x) = (logb)xR∞
0 tx−1dt b−t, x >0, b >1.
19. Using properties of Γ function, prove the following (a) R∞
a e2ax−x2 dx= 12√ πea2, (b) R∞
0
√xe−x3 dx = 1
√π 3 .
20. Prove that the relation B(x, y) = Γ(x)Γ(y)
Γ(x+y) and the following results:
(a) Γ(1) = 1, (b) Γ(12) =√
π.
21. Show that Z π/2
0
sin2n+1θ dθ= Z π/2
0
cos2n+1θ dθ = 22n(n!)2
(2n+ 1)!, n= 0,1,2, . . . . 22. Show that
Γ(x)Γ(x)
2Γ(2x) = 21−2x Z π/2
0
sin2x−1φ dφ.
23. Derive the relation
Γ(n+ 1/2) = (2n)!
22nn!
√π, where n= 0, 1, 2, . . . .
24. Verify the gamma function relation (n+ν 6=−1, −2, −3, . . .) Γ(2n+ 2ν+ 1)
Γ(n+ν+ 1) = 1
√π22n+2vΓ(n+ν+ 1/2).
25. Show that
(a) dzderf(z) = √2πe−z2, (b) R∞
0 e−pxerf(x)dx= 1pep2/4erfc(p/2), p >0, (c) R∞
0 e−px−x2/4 dx=√
πep2erfc(p), p≥0.
26. Use integration by parts to show that (a) R
erfz dz =z erf(z) + √1πe−z2 +C, where C is the constant of integration, (b) R∞
z e−t2 dt= e−z
2
2z − 12R∞ z
e−t2 t2 dt.
27. For any complex number a show thatR∞
0 e−a2x2 dx=
√π 2a.