Laplace Transforms of Periodic Functions Bernd Schr oder logo1 - PowerPoint PPT Presentation

Transforms and New Formulas An Example Double Check Visualization Laplace Transforms of Periodic Functions Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Original DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original ✲ DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ Laplace transform of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform Solution ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Periodic Functions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Periodic Functions 1. A function f is periodic with period T > 0 if and only if for all t we have f ( t + T ) = f ( t ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization Periodic Functions 1. A function f is periodic with period T > 0 if and only if for all t we have f ( t + T ) = f ( t ) . 2. If f is bounded, piecewise continuous and periodic with period T , then � T 1 0 e − st f ( t ) dt � � L f ( t ) = 1 − e − sT logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � � f ( t ) L logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � ∞ 0 e − st f ( t ) dt � � f ( t ) = L logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � ∞ � ( n + 1 ) T ∞ 0 e − st f ( t ) dt = e − st f ( t ) dt � � ∑ f ( t ) = L nT n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � ∞ � ( n + 1 ) T ∞ 0 e − st f ( t ) dt = e − st f ( t ) dt � � ∑ f ( t ) = L nT n = 0 � ( n + 1 ) T ∞ � � e − s ( t − nT )+ nT ∑ = f ( t ) dt nT n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � ∞ � ( n + 1 ) T ∞ 0 e − st f ( t ) dt = e − st f ( t ) dt � � ∑ f ( t ) = L nT n = 0 � ( n + 1 ) T � T ∞ ∞ � � e − s ( t − nT )+ nT 0 e − s ( u + nT ) f ( u ) du ∑ ∑ = f ( t ) dt = nT n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � ∞ � ( n + 1 ) T ∞ 0 e − st f ( t ) dt = e − st f ( t ) dt � � ∑ f ( t ) = L nT n = 0 � ( n + 1 ) T � T ∞ ∞ � � e − s ( t − nT )+ nT 0 e − s ( u + nT ) f ( u ) du ∑ ∑ = f ( t ) dt = nT n = 0 n = 0 � T ∞ e − nsT 0 e − su f ( u ) du ∑ = n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

Transforms and New Formulas An Example Double Check Visualization How Did We Get That? � ∞ � ( n + 1 ) T ∞ 0 e − st f ( t ) dt = e − st f ( t ) dt � � ∑ f ( t ) = L nT n = 0 � ( n + 1 ) T � T ∞ ∞ � � e − s ( t − nT )+ nT 0 e − s ( u + nT ) f ( u ) du ∑ ∑ = f ( t ) dt = nT n = 0 n = 0 � T � � � T ∞ ∞ e − sT � n e − nsT 0 e − su f ( u ) du = 0 e − st f ( t ) dt ∑ ∑ � = n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions