Laplace Transform

Laplace Transform

1. Table

   • Basic
   • s shifting
   • Trigonometric functions
     • Damped oscillations when s shifted
   • Transforms of power
   • Transforms of derivatives
   • Convolution
   • t - shifting

2. Existance of a transform

   • Be intergrable for at least some s
   • Have sub-exponential growth
   • grow rapidly so as to not have a limit
   • Is piecewise continuous
     • continuous except at a finite number of jumps

3. Solving ODE's

   • ODE with t in dom
     • Laplace transform
       • Algebraic equation with s
         • Solve for s
           • Inverse transform
             • Solution in t domain
   • Use derivative transforms

4. Inversion

   • Integral
     • Put to the right of any problem point in F(s)

5. Impulse and delta function

   • Derivative of unit step function

6. Practical applications

   • Circuit theory
     • R.C circuit
   • Mechanical vibrations
   • Mixing two liquids

7. Tools

   • Partial fractions
   • Integration by parts

8. Unit step function

   • u

9. Damped linear systems

   • Transfer Functions
     • Step forcing
     • Impulse forcing
     • Periodic forcing