Transform Pairs
| f(t) | F(s) |
|---|
Partial Fraction Expansion
Enter N(s)/D(s) coefficients (descending order, comma-separated)
N:
D:
Parameters
Decay rate a
1.00
Angular freq. ω
2.00 rad/s
Time range T
5.0 s
—
Poles
—
Zeros
—
DC Gain F(0)
—
Final Value f(∞)
Time Domain f(t)
Frequency Spectrum |F(jω)|
Pole-Zero Map (s-plane)
Theory
$$\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} f(t)e^{-st}\,dt$$Initial Value: $f(0^+) = \lim_{s\to\infty} s F(s)$
Final Value: $\lim_{t\to\infty} f(t) = \lim_{s\to 0} s F(s)$ (poles in LHP)
Partial fractions: $F(s)=\sum_i \dfrac{A_i}{s-p_i}$ → $f(t)=\sum_i A_i e^{p_i t}$
CAE applications: Transfer function analysis · PID steady-state error · Structural dynamics mode decomposition · Impedance design in electrical circuits.