Is it true that Z

HW 2 1.1. Part I: Riemann Improper Integrals.

(1) Do exercises on page 322 in the book by Courant and John volume 1:

5, 6, 8, 9, 10.

(2) Let f ∈ C[1, ∞). Is it true that if Z ∞

1

(f (x))²dx is convergent, then lim

x→∞f (x) = 0.

(3) Let f ∈ C(−∞, ∞). Suppose lim

m→∞

Z m

−m

f (x)dx = 0. Is it true that Z ∞

−∞

f (x)dx = 0?

(4) Let f, g be two nonnegative continuous functions on [1, ∞). Suppose that (a)

Z ∞ 1

g(x)dx is convergent, and (b) lim

x→∞

f (x) g(x) = 0.

Is it true that Z ∞

1

f (x)dx is convergent? If it is true, prove it, if not, give a counterexample.

(5) Let f be a real-valued continuous functions on [1, ∞) and an= f (n), n ≥ 1.

Is it true that lim

n→∞a_n = L if and only if lim

x→∞f (x) = L? If yes, prove it, if not, give a counterexample. Here L ∈ R = R ∪ {±∞}.

(6) (Integral Test) Let f be a nondecreasing positive continuous functions on [1, ∞). Define an= f (n), n ≥ 1.

(a) Show that for each n ≥ 2, a2+ a3+ · · · + an+1≤

Z n+1 1

f (x)dx ≤ a1+ · · · + an.

(Write a mathematical proof of this inequality instead of using pictorial Proof.) Hint:

Since f is nondecreasing on [1, ∞), for each n ≥ 1, f (n + 1) ≤ f (x) ≤ f (n), for x ∈ [n, n + 1].

(b) Prove that (an) and Z ∞

1

f (x)dx both converge or both diverge.

(7) For each n ≥ 1,

an= 1 + 1

2+ · · · + 1 n− ln n.

(a) Show that (an) is decreasing.

(b) Show that an > 0 for all n ≥ 1.

Hint: consider the integral Z n+1

n

dx

x. The limit γ = lim

n→∞an is called the Euler constant.

(8) Test the convergence of the infinite series:

(a)

∞

X

n=3

1/n (ln n)p

ln²n − 1 .

(b)

∞

X

n=1

1 n(1 + ln²n). (c)

∞

X

n=1

8 tan⁻¹n 1 + n² . (9) Evaluate the integrals

(a) Z ∞

0

dx

(1 + x²)(1 + tan⁻¹x).

1

2

(b) Z 2

0

dx p|x − 1|.

(10) Use comparison test, limit comparison test or integration to test the convergence of the following improper integrals.

(a) Z ∞

0

dx 1 + e^x. (b)

Z ∞ 2

√dx x − 1. (c)

Z ∞ 1

e^x xdx.

(d) Z ∞

π

1 + sin x x² dx.

(e) Z ∞

0

e^−5xsin xdx.

(f) Z ∞

0

x³+ 1 x⁴+ 1dx.

(g) Z ∞

0

x ln x (1 + x²)²dx.

(11) Find the values of p ∈ R for which the integral converges : Z ∞

2

dx x(ln x)^p. 1.2. Part II: Laplace Transformation.

(1) Let c > 0 and V be the subset of C[0, ∞) defined by

V = {f ∈ C[0, ∞) : There exists and M > 0 such that |f (t)| ≤ M e^ctfor all t ≥ 0}.

(a) Show that if f, g ∈ V, f + g ∈ V and af ∈ V for a ∈ R. (This implies that V is a R-vector subspace of C[0, ∞)).

(b) Show that V contains β = {1, cos nt, sin nt : n ≥ 1}. (Notice that β is linearly indepen- dent over R. Then V contains a infinite countable R-linearly independent subset and hence V is an infinite dimensional real vector space.)

(c) For each f ∈ V, we define the Laplace transform of f ∈ V by L(f )(s) =

Z ∞ 0

e^−stf (t)dt.

Show that L(f )(s) exists for s > c. In other words, if we set W = C(c, ∞), then L(f ) ∈ W.

(d) Show that L(f + g) = L(f ) + L(g) and that L(af ) = aL(f ). (This implies that L : V → W is a R-linear transformation.)

(2) Let f (t) be a smooth¹function on [0, ∞) whose Laplace transform L(f )(s) exists on [a, ∞) for some a ≥ 0. For s ≥ a, set F (s) = L(f )(s), i.e.,

F (s) = Z ∞

0

e^−stf (t)dt.

Let us denote f^(k)(t) the k-th derivative of f and set f⁽⁰⁾(t) = f (t). Suppose that lim

t→∞f^(k)(t)e^−st= 0, k ≥ 0.

(a) For all n ≥ 1, show that

L(f⁽ⁿ⁾)(s) = sⁿF (s) −X

k=0

s^n−k−1f^(k)(0) by induction.

1In other words, f⁽ⁿ⁾(t) exists and continuous on [0, ∞) for all n ≥ 0.

3

(b) Solve for y = f (t) in the following initial value problem using Laplace transform and the table of Laplace transform.

(i) y⁰+ 3y = 0 with y(0) = 1.

(ii) y⁰⁰− 3y⁰+ 2y = 0 with y(0) = 1 and y⁰(0) = −1.

(iii) y⁰⁰− 2y⁰+ y = 0 with y(0) = 3 and y⁰(0) = 2.

(iv) y⁰⁰+ 4y = 0 with y(0) = 2 and y⁰(0) = −1.

(c) Let F (s) = 2s³− 4s − 8 (s²− s)(s²+ 4).

(i) Find the partial fraction expansion of F (s).

(ii) Solve for f (t) using the table of Laplace transform.

1.3. Part III: Gamma and Beta Function. In the following exercises, you need to use Gamma and Beta functions.

(1) Using Beta function to compute Z ∞

0

e^−x2dx.

(2) Evaluate the following improper integrals.

(a) Z ∞

0

e^−10tt²dt (b)

Z ∞ 0

e^−t2t¹⁰dt.

(3) Let n ≥ 1. Suppose p > 0. Show that 1

Γ(p) Z ∞

0

t^p−1 e^−t− e^−(n+1)t 1 − e^−t

 dt =

n

X

k=1

1 k^p.

Remark. Let us pretend that we can take limit of n → ∞ on the both side of the equation (we may assume p > 1), we obtain

∞

X

n=1

1 n^p = 1

Γ(p) Z ∞

0

t^p−1

 e^−t 1 − e^−t

 dt.

The left hand side of the equation is usually denoted by ζ(p) called the Riemann zeta function.

(4) Suppose that p, q > 0. Show that Z 1

−1

(1 + x)^p−1(1 − x)^q−1dx = 2^p+q−1B(p, q).

(5) Evaluate the following integrals.

(a) Z ∞

0

x³ (1 + x²)¹⁰dx.

(b) Z 1

0

(1 − x²)¹⁰x⁵dx.

(c) Z ∞

0

t¹⁰ (1 + t)¹⁰⁰dt.

(d) Z 3

2

(t − 2)⁵(t − 3)¹⁰dt.

(e) Z π/2

0

sin⁸xdx.

(f) Z π/2

0

cos¹¹x sin¹³xdx.

(g) Z π/2

0

sin¹⁰⁰2xdx.

4

(h) Z 2π

0

sin¹⁰⁰xdx.