Section: 17 | Transforms |

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7 TRANSFORMS

7.1 FOURIER TRANSFORMS

For a piecewise continuous function $F (x)$ over a finite interval $0 \leq x \leq π$ ; the finite Fourier cosine transform of $F (x)$ is $f_{c} (n) = \int_{0}^{π} F (x) cos n x d x (n = 0, 1, 2, \dots)$ If $x$ ranges over the interval $0 \leq x \leq L$ , the substitution $x^{'} = \frac{π x}{L}$ allows the use of this definition, also. The inverse transform is written. $\overset{̲}{F} (x) = \frac{1}{π} f_{c} (0) - \frac{2}{π} \sum_{n = 1}^{x} f_{c} (n) cos n x (0 < x < π)$ where $F (x) = \frac{F (x + ∊) + F (x - ∊)}{2}$ . Note that $F (x +) = F (x -) = F (x)$ at points of continuity. The formula $f_{c}^{(2)} (n) = \int_{0}^{π} F^{''} (x) cos n x d x = - n^{2} f_{c} (n) - F^{'} (0) + (- 1)^{n} F^{'} (π)$ makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform of $F (x)$ is $f_{s} (n) = \int_{0}^{π} F (x) sin n x d x (n = 1, 2, 3, \dots)$ and $\overset{̲}{F} (x) = \frac{2}{π} \sum_{n = 1}^{\infty} f_{s} (n) sin n x (0 < x < π)$ Corresponding to equation (33) we have $f_{s}^{(2)} (n) = \int_{0}^{π} F^{''} (x) sin n x d x = - n^{2} f_{s} (n) - n F (0) - n (- 1)^{n} F (π)$ If $F (x)$ is defined for $x \leq 0$ and is piecewise continuous over any finite interval, and if $\int_{0}^{x} F (x) d x$ is absolutely convergent, then $f_{c} (α) = \sqrt{\frac{2}{π}} \int_{0}^{x} F (x) cos (α x) d x$ is the Fourier cosine transform of $F (x)$ . Furthermore, $\overset{̲}{F} (x) = \sqrt{\frac{2}{π}} \int_{0}^{x} f_{c} (α) cos (α x) d α .$ If $lim_{x \to \infty} d^{n} F / {d x}^{n} = 0$ , then an important property of the Fourier cosine transform is $f_{c}^{(2 r)} (α) = \sqrt{\frac{2}{π}} \int_{0}^{x} (\frac{d^{2 r} F}{{d x}^{2 r}}) cos (α x) d x = - \sqrt{\frac{2}{π}} \sum_{n = 0}^{r - 1} (- 1)^{n} a_{2 r - 2 n - 1} α^{2 n} + (- 1)^{r} α^{2 r} f_{c} (α)$ where $lim_{x \to \infty} d^{r} F / {d x}^{r} = a_{r,}$ which is often useful. Under the same conditions, define the Fourier sine transform of $F (x)$ as follows $f_{s} (α) = \sqrt{\frac{2}{π}} \int_{0}^{x} F (x) sin (α x) d x$ with $\overset{̲}{F} (x) = \sqrt{\frac{2}{π}} \int_{0}^{x} f_{s} (α) sin (α x) d α$ Corresponding to (34) we have $f_{s}^{(2 r)} (α) = \sqrt{\frac{2}{π}} \int_{0}^{\infty} (\frac{d^{2 r} F}{{d x}^{2 r}} sin (α x)) d x = - \sqrt{\frac{2}{π}} \sum_{n = 1}^{r} (- 1)^{n} α^{2 n - 1} a_{2 r - 2 n} + (- 1)^{r - 1} α^{2 r} f_{s} (α)$ Similarly, if $F (x)$ is defined for $- \infty < x < \infty$ , and if $\int_{- \infty}^{\infty} F (x) d x$ is absolutely convergent, then $f (α) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} F (x) e^{i a x} d x$ is the Fourier transform of $F (x)$ , and $\overset{̲}{F} (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (α) e^{- i a x} d α$ Also, if $lim_{| x | \to \infty} | \frac{d^{n} F}{{d x}^{n}} | = 0 (n = 1, 2, \dots, r - 1)$ then $f^{(r)} (α) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} F^{(r)} (x) e^{i α x} d x = (- i α)^{r} f (α)$

7.2 TABLE OF FOURIER COSINE TRANSFORMS

$F (ω) = f_{c} (f) (ω) = \sqrt{\frac{2}{π}} \int_{0}^{\infty} f (x) cos (ω x) d x$ , $ω > 0 .$

No.	$f (x)$	$F (ω)$
1.	${\begin{array}{l} 1 & 0 < x < a \\ 0 & x > a \end{array}$	$\sqrt{\frac{2}{π}} \frac{sin a ω}{ω}$
2.	$x^{p - 1} (0 < p < 1)$	$\sqrt{\frac{2}{π}} \frac{Γ (p)}{ω^{p}} cos \frac{p π}{2}$
3.	${\begin{array}{l} cos x & 0 < x < a \\ 0 & x > a \end{array}$	$\frac{1}{\sqrt{2 π}} (\frac{sin [a (1 - ω)]}{1 - ω} + \frac{sin [a (1 + ω)]}{1 + ω})$
4.	$e^{- x}$	$\sqrt{\frac{2}{π}} \frac{1}{1 + ω^{2}}$
5.	$e^{- x^{2} / 2}$	$e^{- ω^{2} / 2}$
6.	$cos \frac{x^{2}}{2}$	$cos (\frac{ω^{2}}{2} - \frac{π}{4})$
7.	$sin \frac{x^{2}}{2}$	$cos (\frac{ω^{2}}{2} + \frac{π}{4})$

7.3 TABLE OF FINITE COSINE TRANSFORMS

$f_{c} (n) = \int_{0}^{π} F (x) cos n x d x$ , for $n = 0, 1, 2, \dots .$

No.	$f_{c} (n)$	$F (x)$
1.	$(- 1)^{n} f_{c} (n)$	$F (π - x)$
2.	${\begin{array}{l} π & n = 0 \\ 0 & n = 1, 2, \dots \end{array}$	1
3.	${\begin{array}{l} 0 & n = 0 \\ \frac{2}{n} sin \frac{n π}{2} & n = 1, 2, \dots \end{array}$	${\begin{array}{l} 1 & for 0 < x < π / 2 \\ - 1 & for π / 2 < x < π \end{array}$
4.	${\begin{array}{l} \frac{π^{2}}{2} & n = 0 \\ {(- 1)}^{n} - 1 / n^{2} & n = 1, 2 \dots \end{array}$	$x$
5.	${\begin{array}{l} \frac{π^{2}}{6} & n = 0 \\ {(- 1)}^{n} - 1 / n^{2} & n = 1, 2 \dots \end{array}$	$\frac{x^{2}}{2 π}$
6.	$\frac{(- 1)^{n} e^{c} π - 1}{n^{2} + c^{2}}$	$\frac{1}{c} e^{c x}$
7.	$\frac{k}{n^{2} - k^{2}} [(- 1)^{n} cos π k - 1]$ with $k \neq 0, 1, 2, \dots$	$sin k x$
8.	${\begin{array}{l} 0 & m = 1, 2, \dots \\ \frac{{(- 1)}^{n + m} - 1}{n^{2} - m^{2}} & m \neq 1, 2 \dots \end{array}$	$\frac{1}{m} sin m x$
9.	$\frac{1}{n^{2} - k^{2}}$ with $k \neq 0, 1, 2, \dots$	$- \frac{cos k (π - x)}{k sin k π}$
10.	${\begin{array}{l} π / 2 & when n = m \\ 0 & when n \neq m \end{array}$	$cos m x (m = 1, 2, \dots)$

7.4 TABLE OF FOURIER SINE TRANSFORMS

$F (ω) = F_{s} (f) (ω) = \sqrt{\frac{2}{π}} \int_{0}^{\infty} f (x) sin (ω x) d x, ω > 0 .$

No.	$f (x)$	$F (ω)$
1.	${\begin{array}{l} 1 & 0 < x < a \\ 0 & x > a \end{array}$	$\sqrt{\frac{2}{π}} \frac{1 - cos ω a}{ω}$
2.	$x^{p - 1} (0 < p < 1)$	$\sqrt{\frac{2}{π}} \frac{Γ (p)}{ω^{p}} sin \frac{p π}{2}$
3.	${\begin{array}{l} sin x & 0 < x < a \\ 0 & x > a \end{array}$	$\frac{1}{\sqrt{2 π}} (\frac{sin [a (1 - ω)]}{1 - ω} - \frac{sin [a (1 + ω)]}{1 + ω})$
4.	$e^{- x}$	$\sqrt{\frac{2}{π}} \frac{ω}{1 + ω^{2}}$
5.	${x e}^{- x^{2} / 2}$	$ω e^{- ω^{2} / 2}$

7.5 TABLE OF FINITE SINE TRANSFORMS

$f_{s} (n) = \int_{0}^{π} F (x) sin n x d x$ , for $n = 1, 2, \dots$ .

No.	$f_{s} (n)$	$F (x)$
1.	$(- 1)^{n + 1} f_{s} (n)$	$F (π - x)$
2.	$1 / n$	$π - x / π$
3.	$(- 1)^{n + 1} / n$	$x / π$
4.	$1 - (- 1)^{n} / n$	$1$
5.	$\frac{2}{n^{2}} sin \frac{n π}{2}$	${\begin{array}{l} x & when 0 < x < π / 2 \\ π - x & when π / 2 < x < π \end{array}$
6.	$(- 1)^{n + 1} / n^{3}$	$x (π^{2} - x^{2}) / 6 π$
7.	$1 - (- 1)^{n} / n^{3}$	$x (π - x) / 2$
8.	$\frac{π^{2} (- 1)^{n - 1}}{n} - \frac{2 [1 - (- 1)^{n}]}{n^{3}}$	$x^{2}$
9.	$\frac{n}{n^{2} + c^{2}} [1 - (- 1)^{n} e^{c π}]$	$e^{c x}$
10.	$\frac{n}{n^{2} + c^{2}}$	$\frac{sinh c (π - x)}{sinh c π}$
11.	$\frac{n}{n^{2} - k^{2}}$ with $k \neq 0, 1, 2, \dots$	$\frac{sin k (π - x)}{sin k π}$
12.	${\begin{array}{l} π / 2 & when n = m \\ 0 & when n \neq m, m = 1, 2, \dots \end{array}$	$sin m x$
13.	$\frac{n}{n^{2} - k^{2}} [1 - (- 1)^{n} cos k π]$ with $k \neq 1, 2, \dots$ (0 if $n = k$ )	$cos k x$
14.	$\frac{b^{n}}{n}$ with $\| b \| \leq 1$	$\frac{2}{π} arctan \frac{b sin x}{1 - b cos x}$
15.	$\frac{1 - (- 1)^{n}}{n} b^{n}$ with $\| b \| \leq 1$	$\frac{2}{π} arctan \frac{2 b sin x}{1 - b^{2}}$

7.6 TABLE OF FOURIER TRANSFORMS

$F (ω) = F (f) (ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (x) e^{i ω x} d x$

No.	$f (x)$	$F (ω)$
1.	$δ (x)$	$1 / \sqrt{2 π}$
2.	$δ (x - τ)$	$e^{i ω τ} / \sqrt{2 π}$
3.	$δ^{(n)} (x)$	$(- i ω)^{n} / \sqrt{2 π}$
4.	$H (x) = {\begin{array}{l} 1 & x > 0 \\ 0 & x < 0 \end{array}$	$- \frac{1}{i ω \sqrt{2 π}} + \sqrt{\frac{π}{2}} δ (ω)$
5.	$sgn (x) = {\begin{array}{l} 1 & x > 0 \\ - 1 & x < 0 \end{array}$	$- \sqrt{\frac{2}{π}} \frac{1}{i ω}$
6.	${\begin{array}{l} 1 & \| x \| < a \\ - 1 & \| x \| > a \end{array}$	$\sqrt{\frac{2}{π}} \frac{sin a ω}{ω}$
7.	${\begin{array}{l} e Ω t & \| x \| < a \\ 0 & \| x \| > a \end{array}$	$\sqrt{\frac{2}{π}} \frac{sin a (Ω + ω)}{Ω + ω}$
8.	$e^{- a \| x \|} a > 0$	$- \sqrt{\frac{2}{π}} \frac{a}{a^{2} + ω^{2}}$
9.	$\frac{sin a x}{x}$	${\begin{array}{l} \sqrt{\frac{π}{2}} & \| ω \| < a \\ 0 & \| ω \| > a \end{array}$
10.	${\begin{array}{l} e^{i a x} & p < x < q \\ 0 & x < p, x > q \end{array}$	$\frac{i}{\sqrt{2 π}} e^{i p (ω + a)} - e^{i q (ω + a)} / ω + a$
11.	${\begin{array}{l} e^{- c x + i a x} & x > 0 \\ 0 & x < 0 \end{array} (c > 0)$	$\frac{i}{\sqrt{2 π} (ω + a + i c)}$
12.	$e^{- {p x}^{2}} Re p > 0$	$\frac{1}{\sqrt{2 p}} e^{- ω^{2} / 4 p}$
13.	$cos {p x}^{2}$	$\frac{1}{\sqrt{2 p}} cos (\frac{ω^{2}}{4 p} - \frac{π}{4})$
14.	$sin {p x}^{2}$	$\frac{1}{\sqrt{2 p}} cos (\frac{ω^{2}}{4 p} + \frac{π}{4})$
15.	$\| x \|^{- p} (0 < p < 1)$	$\sqrt{\frac{2}{π}} \frac{Γ (1 - p) sin \frac{p π}{2}}{\| ω \|^{1 - p}}$
16.	$e^{- a \| x \|} / \sqrt{\| x \|}$	$\frac{\sqrt{\sqrt{a^{2} + ω^{2}} + a}}{\sqrt{ω^{2} + a^{2}}}$
17.	$\frac{cosh a x}{cosh π x} (- π < a < π)$	$\sqrt{\frac{2}{π}} \frac{cos \frac{a}{2} cosh \frac{ω}{2}}{cos a + cosh ω}$
18.	$\frac{sinh a x}{sinh π x} (- π < a < π)$	$\frac{1}{\sqrt{2 π}} \frac{sin a}{cos a + cosh ω}$
19.	${\begin{array}{l} \frac{1}{\sqrt{a^{2} - x^{2}}} & \| x \| < a \\ 0 & \| x \| > a \end{array}$	$\sqrt{\frac{π}{2}} J_{0} (a ω)$
20.	$\frac{sin [b \sqrt{a^{2} + x^{2}}]}{\sqrt{a^{2} + x^{2}}}$	${\begin{array}{l} 0 & \| ω \| > b \\ \sqrt{\frac{π}{2}} J_{0} (a \sqrt{b^{2} - ω^{2}}) & \| ω \| < b \end{array}$

7.7 TABLE OF FUNCTIONAL RELATIONS FOR FOURIER TRANSFORMS

$F (ω) = F (f) (ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (x) e^{i ω x} d x$

No.	$f (x)$	$F (ω)$
1.	$a g (x) + b h (x)$	$a G (ω) + b H (ω)$
2.	$f (a x) a \neq 0, Im a = 0$	$\frac{1}{\| a \|} F (\frac{ω}{a})$
3.	$f (- x)$	$F (- ω)$
4.	$\overset{̲}{f (x)}$	$\overset{̲}{F (- ω)}$
5.	$f (x - τ) Im τ = 0$	$e^{i ω τ} F (ω)$
6.	$e^{i Ω x} f (x) Im Ω = 0$	$F (ω + Ω)$
7.	$F (x)$	$f (- ω)$
8.	$\frac{d^{n}}{{d x}^{n}} f (x)$	$(- i ω)^{n} F (ω)$
9.	$(i x)^{n} f (x)$	$\frac{d^{n}}{d ω^{n}} F (ω)$

7.8 TABLE OF MULTIDIMENSIONAL FOURIER TRANSFORMS

$F (u) = {(2 π)}^{- n / 2} \int \underset{ℝ^{n}}{\dots} \int f (x) e^{i (x \cdot u)} d x$

No.	$f (x)$	$F (u)$
Two dimensions: let $x = (x, y)$ and $u = (u, v) .$
1.	$f (a x, b y)$	$\frac{1}{\| a b \|} F (\frac{u}{a}, \frac{v}{b})$
2.	$f (x - a, y - b)$	$e^{i (a u + b v)} F (u, v)$
3.	$e^{i (a x + b y)} f (x, y)$	$F (u + a, v + b)$
4.	$F (x, y)$	$(2 π)^{2} F (- u, - v)$
5.	$δ (x - a) δ (y - b)$	$\frac{1}{2 π} e^{- i (a u + b v)}$
6.	$e^{- x^{2} / 4 a - y^{2} / 4 b} a, b > 0$	$2 \sqrt{a b} e^{- {a u}^{2} - {b v}^{2}}$
7.	${\begin{array}{l} 1 & \| x \| < a, \| y \| < b \\ 0 & otherwise \end{array} (rectangle)$	$\frac{2 sin a u sin b v}{π u v}$
8.	${\begin{array}{l} 1 & \| x \| < a \\ 0 & otherwise \end{array} (strip)$	$\frac{2 sin a u}{π u v} δ (v)$
9.	${\begin{array}{l} 1 & x^{2} + y^{2} < a^{2} \\ 0 & otherwise \end{array} (circle)$	$\frac{{a J}_{1} (a \sqrt{u^{2} + v^{2}})}{\sqrt{u^{2} + v^{2}}}$
Three dimensions: let $x = (x, y, z)$ and $u = (u, v, w) .$
10.	$δ (x - a) δ (y - b) δ (z - c)$	$\frac{1}{(2 π)^{3 / 2}} e^{- i (a u + b v + c w)}$
11.	$e^{- x^{2} / 4 a - y^{2} / 4 b - z^{2} / 4 c} a, b, c > 0$	$2^{3 / 2} \sqrt{a b c} e^{- {a u}^{2} - {b v}^{2} - {c w}^{2}}$
12.	${\begin{array}{l} 1 & \| x \| < a, \| y \| < b, \| z \| < c \\ 0 & otherwise \end{array} (box)$	${(\frac{2}{π})}^{3 / 2} \frac{sin a u sin b v sin c w}{u v w}$
13.	${\begin{array}{l} 1 & x^{2} + y^{2} + z^{2} < a^{2} \\ 0 & otherwise \end{array} (ball)$	$\frac{sin a ρ - a ρ cos a ρ}{\sqrt{2 π} ρ^{3}} ρ^{2} = u^{2} + v^{2} + w^{2}$

7.9 TABLE OF LAPLACE TRANSFORMS

$F (s) = L (f) (s) = \int_{0}^{\infty} f (t) e^{- s t} d t .$

No.	$f (t)$	$F (s)$
1.	$δ (t), delta function$	1
2.	$H (t), unit step function or Heaviside function$	$1 / s$
3.	$t$	$1 / s^{2}$
4.	$\frac{t^{n - 1}}{(n - 1)!}$	$1 / s^{n} (n = 1, 2, \dots)$
5.	$1 / \sqrt{π t}$	$1 / \sqrt{s}$
6.	$2 \sqrt{t / π}$	$s^{- 3 / 2}$
7.	$\frac{2^{n} t^{n - 1 / 2}}{\sqrt{π} (2 n - 1)!!}$	$s^{- (n + 1 / 2)} (n = 1, 2, \dots)$
8.	$t^{k - 1}$	$\frac{Γ (k)}{s^{k}} (k > 0)$
9.	$e^{a t}$	$\frac{1}{s - a}$
10.	$t e^{a t}$	$\frac{1}{(s - a)^{2}}$
11.	$\frac{1}{(n - 1)!} t^{n - 1} e^{a t}$	$\frac{1}{(s - a)^{n}} (n = 1, 2, \dots)$
12.	$t^{k - 1} e^{a t}$	$\frac{Γ (k)}{(s - a)^{k}} (k > 0)$
13.	$\frac{1}{a - b} (e^{a t} - e^{b t})$	$\frac{1}{(s - a) (s - b)} (a \neq b)$
14.	$\frac{1}{a - b} ({a e}^{a t} - {b e}^{b t})$	$\frac{s}{(s - a) (s - b)} (a \neq b)$
15.	$- \frac{(b - c) e^{a t} + (c - a) e^{b t} + (a - b) e^{c t}}{(a - b) (b - c) (c - a)}$	$\frac{1}{(s - a) (s - b) (s - c)} (a, b, c distinct)$
16.	$\frac{1}{a} sin a t$	$\frac{1}{s^{2} + a^{2}}$
17.	$cos a t$	$\frac{s}{s^{2} + a^{2}}$
18.	$\frac{1}{a} sinh a t$	$\frac{1}{s^{2} - a^{2}}$
19.	$cosh a t$	$\frac{s}{s^{2} - a^{2}}$
20.	$\frac{1}{a^{2}} (1 - cos a t)$	$\frac{1}{s (s^{2} + a^{2})}$
21.	$\frac{1}{a^{3}} (a t - sin a t)$	$\frac{1}{s^{2} (s^{2} + a^{2})}$
22.	$\frac{1}{2 a^{3}} (sin a t - a t cos a t)$	$\frac{1}{(s^{2} + a^{2})^{2}}$
23.	$\frac{t}{2 a} sin a t$	$\frac{s}{(s^{2} + a^{2})^{2}}$
24.	$\frac{1}{2 a} (sin a t + a t cos a t)$	$\frac{s^{2}}{(s^{2} + a^{2})^{2}}$
25.	$t cos a t$	$\frac{s^{2} - a^{2}}{(s^{2} + a^{2})^{2}}$
26.	$\frac{cos a t - cos b t}{b^{2} - a^{2}}$	$\frac{s}{(s^{2} + a^{2}) (s^{2} + b^{2})} (a^{2} \neq b^{2})$
27.	$\frac{1}{b} e^{a t} sin b t$	$\frac{1}{(s - a)^{2} + b^{2}}$
28.	$e^{a t} cos b t$	$\frac{s - a}{(s - a)^{2} + b^{2}}$
29.	$sin a t cosh a t - cos a t sinh a t$	$\frac{4 a^{3}}{s^{4} + 4 a^{4}}$
30.	$\frac{1}{2 a^{2}} sin a t sinh a t$	$\frac{s}{s^{4} + 4 a^{4}}$
31.	$\frac{1}{2 a^{3}} (sinh a t - sin a t)$	$\frac{1}{s^{4} - a^{4}}$
32.	$\frac{1}{2 a^{2}} (cosh a t - cos a t)$	$\frac{s}{s^{4} - a^{4}}$
33.	$(1 + a^{2} t^{2}) sin a t - cos a t$	$\frac{8 a^{3} s^{2}}{(s^{2} + a^{2})^{3}}$
34.	$\frac{e^{t}}{n!} \frac{d^{n}}{{d t}^{n}} (t^{n} e^{- t})$	$\frac{1}{s} {(\frac{s - 1}{s})}^{n}$
35.	$\frac{1}{\sqrt{π t}} e^{a t} (1 + 2 a t)$	$\frac{s}{(s - a)^{3 / 2}}$
36.	$\frac{1}{2 \sqrt{π t^{3}}} (e^{b t} - e^{a t})$	$\sqrt{s - a} - \sqrt{s - b}$
37.	$\frac{1}{\sqrt{π t}} - {a e}^{a^{2} t} erfc (a \sqrt{t})$	$\frac{1}{\sqrt{s} + a}$
38.	$\frac{1}{\sqrt{π t}} + {a e}^{a^{2} t} erf (a \sqrt{t})$	$\frac{\sqrt{s}}{s - a^{2}}$
39.	$\frac{1}{a} e^{a^{2} t} erf (a \sqrt{t})$	$\frac{1}{\sqrt{s} (s - a^{2})}$
40.	${\begin{array}{l} 0 & when 0 < t < k \\ 1 & when t > k \end{array}$	$\frac{e^{- k s}}{s}$
41.	${\begin{array}{l} 0 & when 0 < t < k \\ t - k & when t > k \end{array}$	$\frac{e^{- k s}}{s^{2}}$
42.	${\begin{array}{l} 0 & when 0 < t < k \\ \frac{{(t - k)}^{p - 1}}{Γ (p)} & when t > k \end{array}$	$\frac{e^{- k s}}{s^{p}}$ (p > 0)
43.	${\begin{array}{l} 1 & when 0 < t < k \\ 0 & when t > k \end{array}$	$\frac{1 - e^{- k s}}{s}$
44.	$\| sin a t \|$	$\frac{a}{s^{2} + a^{2}} coth \frac{π s}{2 a}$
45.	$J_{0} (2 \sqrt{a t})$	$\frac{1}{s} e^{- a / s}$
46.	$\frac{1}{\sqrt{π t}} cos 2 \sqrt{a t}$	$\frac{1}{\sqrt{s}} e^{- a / s}$
47.	$\frac{1}{\sqrt{π t}} cosh 2 \sqrt{a t}$	$\frac{1}{\sqrt{s}} e^{a / s}$
48.	$\frac{a}{2 \sqrt{π t}} e^{- a^{2} / 4 t}$	$e^{- a \sqrt{s}} (a > 0)$
49.	$\frac{1}{\sqrt{π t}} e^{- a^{2} / 4 t}$	$\frac{1}{\sqrt{s}} e^{- a \sqrt{s}} (a \geq 0)$
50.	$Γ^{'} (1) - log t$	$\frac{1}{s} log s$
51.	$t^{k - 1} [\frac{Γ^{'} (k)}{\| Γ (k) \|^{2}} - \frac{log t}{Γ (k)}]$	$\frac{1}{s^{k}} log s (k > 0)$
52.	$e^{a t} [log a - Ei (- a t)]$	$\frac{log s}{s - a} (a > 0)$
53.	$\frac{1}{t} (e^{b t} - e^{a t})$	$log \frac{s - a}{s - b}$
54.	$\frac{2}{t} (1 - cos a t)$	$log \frac{s^{2} + a^{2}}{s^{2}}$
55.	$\frac{2}{t} (1 - cosh a t)$	$log \frac{s^{2} - a^{2}}{s^{2}}$
56.	$\frac{1}{t} sin a t$	$arctan \frac{a}{s}$
57.	$\frac{1}{a \sqrt{π}} e^{- t^{2} / 4 a^{2}}$	$e^{a^{2} s^{2}} erfc (a s) (a > 0)$
58.	$erf (\frac{t}{2 a})$	$\frac{1}{s} e^{a^{2} s^{2}} erfc (a s) (a > 0)$
59.	$\frac{\sqrt{a}}{π \sqrt{t} (t + a)}$	$e^{a s} erfc (\sqrt{a s}) (a > 0)$
60.	$\frac{1}{π t} sin (2 a \sqrt{t})$	$erf (\frac{a}{\sqrt{s}})$

7.10 TABLE OF FUNCTIONAL RELATIONS FOR LAPLACE TRANSFORMS

$F (s) = L (f) (s) = \int_{0}^{\infty} f (t) e^{- s t} d t .$

No.	$f (t)$	$F (s)$
1.	$a f (t) + b g (t)$	$a F (s) + b G (s)$
2.	$f^{'} (t)$	$s F (s) - F (0 +)$
3.	$f^{″} (t)$	$s^{2} F (s) - s F (0 +) - F^{'} (0 +)$
4.	$f^{(n)} (t)$	$s^{n} F (s) - \sum_{k = 0}^{n - 1} s^{n - 1 - k} F^{(k)} (0 +)$
5.	$\int_{0}^{t} f (τ) d τ$	$\frac{1}{s} F (s)$
6.	$\int_{0}^{t} \int_{0}^{τ} f (u) d u d τ$	$\frac{1}{s^{2}} F (s)$
7.	$\int_{0}^{t} f_{1} (t - τ) f_{2} (τ) d τ = f_{1} * f_{2}$	$F_{1} (s) F_{2} (s)$
8.	$t f (t)$	$- F^{'} (s)$
9.	$t^{n} f (t)$	$(- 1)^{n} F^{(n)} (s)$
10.	$\frac{1}{t} f (t)$	$\int_{s}^{\infty} F (z) d z$
11.	$e^{a t} f (t)$	$F (s - a)$
12.	$f (t - b) with f (t) = 0 for t < 0$	$e^{- b s} F (s)$
13.	$\frac{1}{c} f (\frac{t}{c})$	$F (c s)$
14.	$\frac{1}{c} e^{b t / c} f (\frac{t}{c})$	$F (c s - b)$
15.	$f (t + a) = f (t)$	$\int_{0}^{a} e^{- s t} f (t) d t / 1 - e^{- a s}$
16.	$f (t + a) = - f (t)$	$\int_{0}^{a} e^{- s t} f (t) d t / 1 + e^{- a s}$

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Cookie Policy

We have developed this cookie policy (the “Cookie Policy”) in order to explain how we use cookies and similar technologies (together, “Cookies”) on this website (the “Website”) and to demonstrate our firm commitment to the privacy of your personal information.

The first time that you visit our Website, we notify you about our use of Cookies through a notification banner. By continuing to use the Website, you consent to our use of Cookies as described in this Cookie Policy. However, you can choose whether or not to continue accepting Cookies at any later time. Information on how to manage Cookies is set out later in this Cookie Policy.

Please note that our use of any personal information we collect about you is subject to our Privacy Policy.

What are Cookies?

Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.

Cookies can be ‘Session Cookies’ or ‘Persistent Cookies’. Session Cookies allow a website to link a series of your actions during one browser session, for example to remember the items you have added to a shopping basket. Session Cookies expire after a browser session and are therefore not stored on your computer or device afterwards. Persistent Cookies are stored on your computer or device between browser sessions and can be used when you make subsequent visits to the website, for example to remember your website preferences, such as language or font size.

Cookies We Use and Their Purpose

We use three types of Cookies - ‘Strictly Necessary’ Cookies, ‘Performance’ Cookies and ‘Functionality’ Cookies. Each type of Cookie and the purposes for which we use them are described in this section. To learn about the specific Cookies we use, please see our List of Cookies.

1. Strictly Necessary Cookies

‘Strictly Necessary’ Cookies enable you to move around the Website and use essential features. For example, if you log into the Website, we use a Cookie to keep you logged in and allow you to access restricted areas, without you having to repeatedly enter your login details. If you are registering for or purchasing a product or service, we will use Cookies to remember your information and selections, as you move through the registration or purchase process.

Strictly Necessary Cookies are necessary for our Website to provide you with a full service. If you disable them, certain essential features of the Website will not be available to you and the performance of the Website will be impeded.

2. Performance Cookies

‘Performance’ Cookies collect information about how you use our Website, for example which pages you visit and if you experience any errors. These Cookies don’t collect any information that could identify you – all the information collected is anonymous. We may use these Cookies to help us understand how you use the Website and assess how well the Website performs and how it could be improved.

3. Functionality Cookies

‘Functionality’ Cookies enable a website to provide you with specific services or a customised experience. We may use these Cookies to provide you with services such as watching a video or adding user comments. We may also use such Cookies to remember changes you make to your settings or preferences (for example, changes to text size or your choice of language or region) or offer you time-saving or personalised features.

You can control whether or not Functionality Cookies are used, but disabling them may mean we are unable to provide you with some services or features of the Website.

First and Third Party Cookies

The Cookies placed on your computer or device include ‘First Party’ Cookies, meaning Cookies that are placed there by us, or by third party service providers acting on our behalf. Where such Cookies are being managed by third parties, we only allow the third parties to use the Cookies for our purposes, as described in this Cookie Policy, and not for their own purposes.

The Cookies placed on your computer or device may also include ‘Third Party’ Cookies, meaning Cookies that are placed there by third parties. These Cookies may include third party advertisers who display adverts on our Website and/or social network providers who provide ‘like’ or ‘share’ capabilities (see the above section on Targeting or Advertising Cookies). They may also include third parties who provide video content which is embedded on our Website (such as YouTube). Please see the website terms and policies of these third parties for further information on their use of Cookies.

To learn about the specific First Party and Third Party Cookies used by our, please see our List of Cookies.

Managing Cookies

You always have a choice over whether or not to accept Cookies. When you first visit the Website and we notify you about our use of Cookies, you can choose not to consent to such use. If you continue to use the Website, you are consenting to our use of Cookies for the time being. However, you can choose not to continue accepting Cookies at any later time. In this section, we describe ways to manage Cookies, including how to disable them.

You can manage Cookies through the settings of your internet browser. You can choose to block or restrict Cookies from being placed on your computer or device. You can also review periodically review the Cookies that have been placed there and disable some or all of them.

You can learn more about how to manage Cookies on the following websites: www.allaboutcookies.org and www.youronlinechoices.com.

Please be aware that if you choose not to accept certain Cookies, it may mean we are unable to provide you with some services or features of the Website.

Changes to Cookie Policy

In order to keep up with changing legislation and best practice, we may revise this Cookie Policy at any time without notice by posting a revised version on this Website. Please check back periodically so that you are aware of any changes.

Questions or Concerns

If you have any questions or concerns about this Cookie Policy or our use of Cookies on the Website, please contact us by email to [email protected]

You can also contact the Privacy Officer for the Informa PLC group at [email protected].