Math Problems Vault

Problem 3835 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.3 "Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals"

For the function $f(t)$ , find the Laplace transform

F(p)=\int _{0}^{+\infty }e^{-pt}f(t)\, dt \quad (p>0).

Let:

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(a)

$f(t)=t^{n}$ ( $n$ a natural number);

(b)

$f(t)=\sqrt{t}$ ;

(c)

$f(t)=e^{at}$ ;

(d)

$f(t)=te^{-at}$ ;

(e)

$f(t)=\cos t$ ;

(f)

$f(t)=\dfrac {1-e^{-t}}{t}$ ;

(g)

$f(t)=\sin a\sqrt{t}$ .

Problem 3836 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.3 "Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals"

Prove the formula (the Lipschitz integral)

\int _{0}^{+\infty }e^{-at}J_{0}(bt)\, dt=\dfrac {1}{\sqrt{a^{2}+b^{2}}} \quad (a>0),

where $J_{0}(x)=\dfrac {1}{\pi }\displaystyle \int_{0}^{\pi }\cos (x\sin \varphi )\, d\varphi$ is the Bessel function of order 0 (see problem 3726: $J_{0}(x)=\dfrac {1}{\pi }\displaystyle \int_{0}^{\pi }\cos (x\sin \varphi )\, d\varphi$ is a solution of the Bessel equation $xy''+y'+xy=0$ ).

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Problem 2.142 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Singular Integrals. Weierstrass' Approximation Theorem"

Let the function $\varphi (x)$ be defined and continuous for $x \geq 0$ . Suppose that the integral

J(k) = \int _{0}^{\infty } e^{-kx} \varphi (x) \, dx

converges for $k = k_{0}$ and that it vanishes for a sequence of $k$ 's increasing in arithmetic progression:

J(k_{0}) = J(k_{0}+\alpha ) = J(k_{0}+2\alpha ) = \cdots = J(k_{0}+n\alpha ) = \cdots = 0, \qquad \alpha > 0 .

Then $\varphi (x)$ vanishes identically.

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Problem 5.80 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6 "Laguerre's Proof of Descartes' Rule of Signs"

Denote the number of reversals of sign of the function $\varphi (\lambda )$ in the interval $0 < \lambda < \infty$ by $R$ , and the number of real zeros of the integral

F(x) = \int _0^{\infty } \varphi (\lambda ) e^{-\lambda x}\, d\lambda

by $Z$ . Then we have $Z \leq R$ .

In the number of zeros $Z$ we naturally include only the zeros lying in the interior of the region of convergence (a half-plane bounded on the left). (Proof not by application of a limiting process to 77 but by appropriate translation of the proof of 77 to the present case!)

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Problem 5.82 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6 "Laguerre's Proof of Descartes' Rule of Signs"

The number of positive zeros lying in the interior of the region of convergence does not exceed the number of reversals of sign of the function $\Phi (\lambda ) = \int_0^{\lambda } \varphi (x)\, dx$ . (Analogue of 44.)

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Problem 2.15.22 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.15 "Zeros of Holomorphic Functions"

Put $f_{n}(t) = t^{n} e^{-t}(0 \leq t < \infty , n = 0,1,2, \ldots )$ and prove that the set of all finite linear combinations of the functions $f_{n}$ is dense in $L^{2}(0, \infty )$ . Hint: If $g \in L^{2}(0, \infty )$ is orthogonal to each $f_{n}$ and if

F(z) = \int _{0}^{\infty } e^{-t z} \overline{g(t)} d t \quad (\mathrm{Re}\left(z\right) > 0)

then all derivatives of $F$ are 0 at $z = 1$ . Consider $F(1+i y)$ .

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Problem 8.7 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 8 "Sequences and Series of Regular Functions. Integrals Depending on a Parameter."

Let the function $\varphi (t)$ be continuous for $t \geqslant 0$ and satisfy the condition

\varlimsup _{t \rightarrow +\infty } \frac{\ln \left|\varphi (t)\right|}{t} = \sigma .

Prove that the integral

\int _{0}^{\infty } \varphi (t) e^{-t z} d t

represents a function regular in the half-plane $\mathrm{Re}\left(z\right) > \sigma$ .

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Problem 32.1 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 32 "The Laplace Transform (Operational Calculus) and its Application to Solving Differential Equations"

(1)(2)(3)(4)

Find the transform $F(p)$ of the original $f(t)$ , if:

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(1)

$f(t) = \sin t \sin 3 t$ ;

(2)

$f(t) = t \operatorname {sh} t ;$

(3)

$f(t) = \sin^{3} t$ ;

(4)

$f(t) = \cos^{4} t+\sin^{4} t$ .

Problem 32.2 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 32 "The Laplace Transform (Operational Calculus) and its Application to Solving Differential Equations"

Find the original $f(t)$ from its transform $F(p)$ , if:

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(1)

$F(p) = \frac{1}{p^{2}-4 p+5}$ ;

(2)

$F(p) = \frac{1}{p^{2}\left(p^{2}+1\right)}$ ;

(3)

$F(p) = \frac{p+1}{p^{3}-2 p^{2}-5 p+6}$ ;

(4)

$F(p) = \frac{1}{p(p-1)\left(p^{2}+4\right)}$ ;

(5)

$F(p) = \frac{1}{(p+1)(p+3)^{3}}$ ;

(6)

$F(p) = \frac{p^{2}+2 p-1}{p^{3}+3 p^{2}+3 p+1}$ ;

(7)

$F(p) = \frac{(5 p+3)}{(p-1)\left(p^{2}+2 p+5\right)}$ ;

(8)

$F(p) = \frac{p^{2}}{\left(p^{2}+4\right)\left(p^{2}+9\right)}$ ;

(9)

$F(p) = \frac{1}{(p+1)^{3}(p+3)}$ ;

(10)

$F(p) = \frac{1}{(p-1)^{2}(p-2)^{3}}$ .

Problem 32.3 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 32 "The Laplace Transform (Operational Calculus) and its Application to Solving Differential Equations"

Solve the Cauchy problem for the differential equation with the given initial conditions:

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(1)

$x^{\prime \prime }(t)-2 x^{\prime }(t)+x(t) = t^{2} e^{t}, x(0) = 0, x^{\prime }(0) = 1$ ;

(2)

$x^{(3)}(t)+3 x^{(2)}(t)+3 x^{\prime }(t)+x(t) = 1, x(0) = x^{\prime }(0) = x^{\prime \prime }(0) = 0$ ;

(3)

$x^{(3)}(t)-2 x^{(2)}(t)+x^{\prime }(t) = 4, x(0) = 1, x^{\prime }(0) = 2, x^{\prime \prime }(0) = -2$ ;

(4)

$x^{(3)}(t)+x^{(2)}(t) = \cos t, x(0) = -2, x^{\prime }(0) = x^{\prime \prime }(0) = 0$ ;

(5)

$x^{\prime \prime }(t)+x(t) = 2 \cos t, x(0) = -1, x^{\prime }(0) = 1$ ;

(6)

$x^{\prime \prime }(t)+4 x(t) = 2 \sin 2 t, x(0) = -1, x^{\prime }(0) = 0$ ;

(7)

$x^{(4)}(t)-x(t) = 1, x(0) = 1, x^{\prime }(0) = 1, x^{\prime \prime }(0) = 2, x^{(3)}(0) = -1$ ;

(8)

$x^{(4)}(t)-5 x^{(2)}(t)+10 x^{\prime }(t)-6 x(t) = 0$ , $x(0) = 1, x^{\prime }(0) = 0, x^{\prime \prime }(0) = 6, x^{(3)}(0) = -14$ .

Problem 32.4 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 32 "The Laplace Transform (Operational Calculus) and its Application to Solving Differential Equations"

(1)(2)(3)(4)

Solve the Cauchy problem for the system of differential equations with the given initial conditions:

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(1)

$\begin{cases} x^{\prime }(t) = -x(t)+2 y(t), \\ y^{\prime }(t) = -x(t)-4 y(t), \end{cases} \quad x(0) = y(0) = 1 ;$

(2)

$\begin{cases} x^{\prime }(t)-2 x(t)-4 y(t) = 0, \\ y^{\prime }(t)+x(t)+2 y(t) = 0, \end{cases} \quad x(0) = 2, \quad y(0) = 1$ ;

(3)

$\begin{cases} x^{\prime }(t)+3 x(t)-4 y(t) = 9 e^{2 t}, \\ 2 x(t)+y^{\prime }(t)-3 y(t) = 3 e^{2 t}, \end{cases} \quad x(0) = 2, \quad y(0) = 0$ ;

(4)

$\begin{cases} x^{\prime }(t)+y^{\prime }(t) = 1, \\ x^{(2)}(t)+y^{\prime }(t)+x(t) = e^{t}, \end{cases} x(0) = 1, x^{\prime }(0) = 2, y(0) = 0, y^{\prime }(0) = -1$ .

Problem 23.1.15 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 23.1 "Chapter Problems"

(i)(ii)

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(i)

Let $\overrightarrow {u} = (u_{1}, \ldots , u_{n})^{T}$ and $A$ be an $n \times n$ matrix over $\mathbb {R}$ . Solve the initial value problem of the system of linear differential equations

\frac{d\overrightarrow {u}}{dt} = A\overrightarrow {u}

using the Laplace transform, where $\overrightarrow {u}(t=0) = \overrightarrow {u}(0)$ . Note that (applying integration by parts)

\mathcal{L}(d\overrightarrow {u}/dt)(s) = \int _{0}^{\infty } e^{-st}\frac{d\overrightarrow {u}}{dt}\, dt = e^{-st}\overrightarrow {u}(t)\Big|_{t=0}^{t=\infty } + s\int _{0}^{\infty } e^{-st}\overrightarrow {u}(t)\, dt = s\overrightarrow {U}(s) - \overrightarrow {u}(0) .

(ii)

Apply it to the $2 \times 2$ Hadamard matrix

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} .

Problem 25.1.9 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 25.1 "Chapter Problems"

Let $A$ be an $n \times n$ matrix. Let $\omega , \mu \in \mathbb {R}$ . Assume that

\left\| e^{tA}\right\| \leq M e^{\omega t}, \qquad t \geq 0

and $\mu > \omega$ . Then we have

(\mu I_{n} - A)^{-1} \equiv \int _{0}^{\infty } e^{-\mu t} e^{tA} \, dt. \tag {1}

Calculate the left- and right-hand side of (1) for the matrix $A = \sigma_{1}$ .

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References