Engineering Mathematics III: Laplace Transform and Periodic Functions
Explore the concept of Laplace Transform, a powerful tool for solving linear differential equations, and delve into periodic functions in Engineering Mathematics III. Understand definitions, properties, and elementary transformations along with related resources for further learning.
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18mat31 18mat31 Module 1 Module 1 Engineering Mathematics - III MANOHAR KUMAR K.N MANOHAR KUMAR K.N Assistant Professor Department of Mathematics K.S. School of Engineering and Management Bengaluru - 560109
LAPLACE TRANSFORM Laplace Transform is an integral transform which is developed by Mathematician Pierre-Simon Laplace. Laplace transform method is a powerful tool or technique for solving Linear differential equation with initial condition. Generally transform means To change one form to another form without changing its value. Laplace transform is a simple way of converting functions in time domain to the function of frequency domain
Definition: Let f(t) be a function of real variable t defined for t 0. Laplace transform of f(t) is denoted by L[f t ] and is defined by ? ?? ? ? ?? L f t = 0 Where s is a parameter, real or complex. Properties of Laplace Transform: Linearity Property L a f t + bg t = aL f t + bL[g t ] First Shifting Property = F s then L eat f t If L f t = F(s a) Change of Scale Property =1 ? ? If L f t = F s then L f ?t a F
Elementary Transformation: L a = a s 1 L eat= s a 1 L e at= s+a a L sinat = s2+a2 s L cosat = s2+a2 a L sinhat = s2 a2 s s2 a2 L coshat = n! L tn= sn+1
Periodic Function: Definition: A function ?(?) is said to be periodic function of period ? > 0 if ? ? + ?? = ? ? where n = 1,2,3 Example: ???? and ???? are periodic functions of period 2 Statement: If ?(?) is a periodic function of period T then ? 1 ? ?? ? ? ?? ?[? ? = 1 ? ?? 0
Unit step function(Heaviside function): Definition: The unit step function ?(? ?) is defined as follows. 0 , ? ? 1 , ? > ? where ? is positive constant. ? ? ? = Properties: ? ?? ? 1. ?[? ? ? = 2. ?[? ? ? ? ? ? = ? ?? ?(?) ?1? , ?2? , ? ? ? > ? 3. ?? ? ? = ? ?? ? ? = ?1? + ?2? ?1? ? ? ?
Related Links: 1. https://www.khanacademy.org/math/differential-equations/laplace- transform/laplace-transform-tutorial/v/laplace-transform-1 2. https://www.youtube.com/watch?v=Lmj3s4S62fQ 3. https://www.youtube.com/watch?v=RDtITuZDZi4 4. https://www.youtube.com/watch?v=c9NibpoQjDk 5. https://www.youtube.com/watch?v=9RJml41PFnc
