📐KIX1002: Engineering Mathematics II/
Laplace Table

Laplace Table

PDF Version

Laplace_Table.pdf
125.5KB

LaTeX source in Overleaf

www.overleaf.com
An online LaTeX editor that's easy to use. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more.
https://www.overleaf.com/read/tcpcxgryrxns
f(t)=L−1{F(s)}f(t)=\mathcal{L}^{-1}\{F(s)\}
F(s)=L{f(t)}F(s)=\mathcal{L}\{f(t)\}
1.
11
1s\dfrac{1}{s}
2.
eat\mathbf{e}^{a t}
1s−a\dfrac{1}{s-a}
3.
tn,n=1,2,3,…t^{n}, \quad n=1,2,3, \ldots
n!sn+1\dfrac{n !}{s^{n+1}}
4.
tp,p>−1t^{p}, p>-1
Γ(p+1)sp+1\dfrac{\Gamma(p+1)}{s^{p+1}}
5.
t\sqrt{t}
π2s32\dfrac{\sqrt{\pi}}{2 s^{\frac{3}{2}}}
6.
tn−12, n=1,2,3,…t^{n-\frac{1}{2}},~ n=1,2,3, \ldots
1⋅3⋅5⋯(2n−1)π2nsn+12\dfrac{1 \cdot 3 \cdot 5 \cdots(2 n-1) \sqrt{\pi}}{2^{n} s^{n+\frac{1}{2}}}
7.
sin⁡(at)\sin (a t)
as2+a2\dfrac{a}{s^{2}+a^{2}}
8.
cos⁡(at)\cos (a t)
ss2+a2\dfrac{s}{s^{2}+a^{2}}
9.
tsin⁡(at)t \sin (a t)
2as(s2+a2)2\dfrac{2 a s}{\left(s^{2}+a^{2}\right)^{2}}
10.
tcos⁡(at)t \cos (a t)
s2−a2(s2+a2)2\dfrac{s^{2}-a^{2}}{\left(s^{2}+a^{2}\right)^{2}}
11.
sin⁡(at)−atcos⁡(at)\sin (a t)-a t \cos (a t)
2a3(s2+a2)2\dfrac{2 a^{3}}{\left(s^{2}+a^{2}\right)^{2}}
12.
sin⁡(at)+atcos⁡(at)\sin (a t)+a t \cos (a t)
2as2(s2+a2)2\dfrac{2 a s^{2}}{\left(s^{2}+a^{2}\right)^{2}}
13.
cos⁡(at)−atsin⁡(at)\cos (a t)-a t \sin (a t)
s(s2−a2)(s2+a2)2\dfrac{s\left(s^{2}-a^{2}\right)}{\left(s^{2}+a^{2}\right)^{2}}
14.
cos⁡(at)+atsin⁡(at)\cos (a t)+a t \sin (a t)
s(s2+3a2)(s2+a2)2\dfrac{s\left(s^{2}+3 a^{2}\right)}{\left(s^{2}+a^{2}\right)^{2}}
15.
sin⁡(at+b)\sin (a t+b)
ssin⁡(b)+acos⁡(b)s2+a2\dfrac{s \sin (b)+a \cos (b)}{s^{2}+a^{2}}
16.
cos⁡(at+b)\cos (a t+b)
scos⁡(b)−asin⁡(b)s2+a2\dfrac{s \cos (b)-a \sin (b)}{s^{2}+a^{2}}
17.
sinh⁡(at)\sinh (a t)
as2−a2\dfrac{a}{s^{2}-a^{2}}
18.
cosh⁡(at)\cosh (a t)
ss2−a2\dfrac{s}{s^{2}-a^{2}}
19.
eatsin⁡(bt)\mathbf{e}^{a t} \sin (b t)
b(s−a)2+b2\dfrac{b}{(s-a)^{2}+b^{2}}
20.
eatcos⁡(bt)\mathbf{e}^{a t} \cos (b t)
s−a(s−a)2+b2\dfrac{s-a}{(s-a)^{2}+b^{2}}
21.
eatsinh⁡(bt)\mathbf{e}^{a t} \sinh (b t)
b(s−a)2−b2\dfrac{b}{(s-a)^{2}-b^{2}}
22.
eatcosh⁡(bt)\mathbf{e}^{a t} \cosh (b t)
s−a(s−a)2−b2\dfrac{s-a}{(s-a)^{2}-b^{2}}
23.
tneat,  n=1,2,3,…t^{n} \mathbf{e}^{a t},~~ n=1,2,3, \ldots
n!(s−a)n+1\dfrac{n !}{(s-a)^{n+1}}
24.
f(ct)f(c t)
1cF(sc)\dfrac{1}{c} F\left(\frac{s}{c}\right)
25.
uc(t)=u(t−c)u_{c}(t)=u(t-c)
e−css\dfrac{\mathbf{e}^{-c s}}{s}
26.
δ(t−c)\delta(t-c)
e−cs\mathbf{e}^{-c s}
27.
uc(t)f(t−c)u_{c}(t) f(t-c)
e−csF(s)\mathbf{e}^{-c s} F(s)
28.
uc(t)g(t)u_{c}(t) g(t)
e−csL{g(t+c)}\mathbf{e}^{-c s} \mathcal{L}\{g(t+c)\}
29.
ectf(t)\mathbf{e}^{c t} f(t)
F(s−c)F(s-c)
30.
tnf(t), n=1,2,3,…t^{n} f(t), ~ n=1,2,3, \ldots
(−1)nF(n)(s)(-1)^{n} F^{(n)}(s)
31.
1tf(t)\dfrac{1}{t} f(t)
∫s∞F(u)du\int_{s}^{\infty} F(u) d u
32.
∫0tf(v)dv\int_{0}^{t} f(v) d v
F(s)s\dfrac{F(s)}{s}
33.
∫0tf(t−τ)g(τ)dτ\int_{0}^{t} f(t-\tau) g(\tau) d \tau
F(s)G(s)F(s) G(s)
34.
f(t+T)=f(t)f(t+T)=f(t)
∫0Te−stf(t)dt1−e−sT\dfrac{\int_{0}^{T} \mathbf{e}^{-s t} f(t) d t}{1-\mathbf{e}^{-s T}}
35.
f′(t)f^{\prime}(t)
sF(s)−f(0)s F(s)-f(0)
36.
f′′(t)f^{\prime \prime}(t)
s2F(s)−sf(0)−f′(0)s^{2} F(s)-s f(0)-f^{\prime}(0)
37.
f(n)(t)f^{(n)}(t)
snF(s)−sn−1f(0)−sn−2f′(0)⋯−sf(n−2)(0)−f(n−1)(0)s^{n} F(s)-s^{n-1} f(0)-s^{n-2} f^{\prime}(0) \cdots-s f^{(n-2)}(0)-f^{(n-1)}(0)