What is the Laplace transform of f(t)?
Type a function of t, such as t^3 e^(4t) or sin(2t). The page gives its Laplace transform F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt.
- F(s) =
- 6/(s - 4)^4
The Laplace transform of t^3 e^(4t) is F(s) = 6/(s - 4)^4.
F(s) =: 6/(s - 4)^4. The Laplace transform of t^3 e^(4t) is F(s) = 6/(s - 4)^4.
How to calculate
Finds the Laplace transform F(s) of a function f(t), term by term, as partial fractions, checked by numeric integration.
Example with the default inputs (Function f(t) t^3 e^(4t)): The Laplace transform of t^3 e^(4t) is F(s) = 6/(s - 4)^4.
Method: F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt. A computer algebra system transforms each term; each transform is checked against numeric integration at three values of s.
- f(t) is defined for t ≥ 0.
- An answer that fails its check is not shown.
Worked examples
Each example is checked against the calculator on every build.
- Function f(t) t^3 e^(4t) gives F(s) = 6/(s - 4)^4.Source: Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.5. https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_Transform
- Function f(t) sin(2t) gives F(s) = 2/(4 + s^2).Source: Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.4 (ω = 2: ω/(s² + ω²))
- Function f(t) cosh(3t) gives F(s) = 1/(2 (s - 3)) + 1/(2 (3 + s)).Source: Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.6 (b = 3: s/(s² − 9), in partial fractions)
How it works
The Laplace transform of f(t) is
F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt
A computer algebra system (nerdamer, open source) does the algebra, in the background after you start typing.
- Term by term. f is split into its top-level terms (the parts joined by + and −, each with its sign), because the transform is linear.
- Each term T is transformed by the algebra. Its answer is checked: it must equal a numeric integral of e^(−st) T(t) (from t = 10⁻¹² to 60/s) at s = 4, 6 and 8, to 1 part in 10⁵. If that fails, the algebra transforms e^(−10t) T(t) instead (checked the same way), and s is replaced by s − 10 in the result: by the shift rule L[e^(ct) T] = F(s − c), this is the transform of T. The check then runs at s = 14, 16 and 18 for T, where the integral converges fast.
- One fraction. The sum of the transforms is written as one fraction p/q. First every power u^k with a whole-number k whose base u is negative at s = 10⁶ (such as (3 − s)^3) is written as ±(−u)^k (+ for even k, − for odd); then, if q is negative at s = 10⁶, both p and q change sign. This keeps the algebra from losing a sign.
- Partial fractions. The algebra writes p/q as partial fractions, checked equal to it at 20 points (to 1 part in 10⁹). If it cannot, the page shows the sum of the terms' transforms as it is.
The page shows F(s) = in the syntax you type (s as the variable, ^ for powers). A result that holds a number the algebra could only give rounded (a whole number past 2⁵³; a fraction p/q, p and q being the numbers that multiply its top and its bottom, such as 288557167/(342919925e), when q after removing its factors 2 and 5 is over 1,000,000, or is over 1 while |p| times it is over 10¹²; or a decimal with more than 12 significant digits) is not shown.
What you can type
- A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
- f(t) uses the variable t. Numbers can have decimals; + − * / and ^ for powers; brackets; a number or bracket next to t multiplies (3t, 2(t + 1), t e^(2t)).
- Constants pi and e; functions exp, sqrt, sin, cos, sinh, cosh and the others of the integral calculator. Angles are in radians.
What gets no answer
- A function with no Laplace transform (1/t), or one the algebra cannot transform (unit steps and delta functions cannot be typed).
- A term growing like e^(ct) with c of about 14 or more: its check fails even after the shift.
- A step that fails its check, finds no formula, or takes over 3 seconds.
Worked examples by hand
f(t) = t³e^(4t) (Trench, section 8.1, Example 8.1.5). L[t³] = 3!/s⁴ = 6/s⁴, and the shift rule replaces s by s − 4: F(s) = 6/(s − 4)^4, for s > 4.
f(t) = sin(2t) (Trench, section 8.1, Example 8.1.4, with ω = 2). L[sin(ωt)] = ω/(s² + ω²), so F(s) = 2/(4 + s^2), for s > 0.
f(t) = cosh(3t) (Trench, section 8.1, Example 8.1.6, with b = 3). L[cosh(bt)] = s/(s² − b²) = s/(s² − 9), for s > 3. In partial fractions, s/((s − 3)(s + 3)) = (1/2)/(s − 3) + (1/2)/(s + 3), written 1/(2 (s − 3)) + 1/(2 (3 + s)).
Other questions people ask
What is the Laplace transform?
It turns a function of time f(t), for t ≥ 0, into a function of s: F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt. Derivatives in t become multiplication by s, which turns linear differential equations into algebra.
What are the most common transforms?
L[1] = 1/s, L[tⁿ] = n!/s^(n+1), L[e^(at)] = 1/(s − a), L[sin(bt)] = b/(s² + b²), L[cos(bt)] = s/(s² + b²), L[tⁿ e^(at)] = n!/(s − a)^(n+1), and L[e^(at) sin(bt)] = b/((s − a)² + b²). Each holds for s larger than the growth rate of f.
How does the page handle a sum like 3t⁴ − 2t + 5?
The transform is linear, so it transforms each term and adds: 3 · 24/s⁵ − 2/s² + 5/s = 72/s⁵ − 2/s² + 5/s.
For which s does F(s) hold?
For s larger than the exponential growth rate of f: s > 0 for 1, t and sin(2t); s > 4 for t³e^(4t). The page gives the formula; the integral diverges for smaller s.
Why is there no transform of 1/t?
The integral ∫ from 0 of e^(−st)/t dt is infinite near t = 0 for every s, so 1/t has no Laplace transform. Functions that grow faster than any exponential, such as e^(t²), have none either.
How is the answer checked?
Each term's transform is compared with a numeric integral of e^(−st) f(t) at three values of s. The partial-fraction form is compared with the sum it rewrites at 20 points. If a check fails, the page says "No verified answer".