Laplace transform

• How to Calculate the Output of a Transfer Function by Hand
  Calculate an LTI system’s output by multiplying its transfer function by the input’s Laplace transform, then taking the inverse Laplace transform.
• What Is the Laplace Transform of a Trigonometric Function?
  The Laplace transforms of sine and cosine are simple rational functions; learn the formulas, convergence conditions, exponential shifts, and a worked example.
• How to Find the Laplace Transform of a Mass-Spring-Damper System
  Transform m x'' + c x' + kx = F(t) into the displacement formula, including initial conditions, the transfer function, and a worked step-force example.
• How to Find a New Output When the Input Changes
  For a linear time-invariant system, find the new output by multiplying its transfer function by the new input transform, then invert; account for initial conditions.
• Laplace Transforms of (t − 2)u(t − 2) and tu(t − 2) − 2u(t − 2)
  Both expressions are identical, and the second shifting theorem gives their Laplace transform as e^(−2s)/s² for Re(s) > 0.
• How to Find the Inverse Laplace Transform of 1/[s²(s² + ω²)]
  Find the inverse Laplace transform of 1/[s²(s² + ω²)] by partial fractions, and check the result and the special case ω = 0.
• How to Find the Inverse Laplace Transform of arctan(2/s²)
  Expand arctan(2/s²) as a power series and invert term by term to obtain an exact, convergent series for its inverse Laplace transform.
• What Basics Do You Need to Study the Laplace Transform?
  Learn the calculus, algebra, and differential-equation basics for Laplace transforms, with a worked example showing how the method solves an ODE.
• What Is the Laplace Transform of $t e^{at}\sin(bt)$?
  The Laplace transform of t e^{at} sin(bt) is 2b(s−a)/([(s−a)^2+b^2]^2), for real a, nonzero b, and Re(s)>a.