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- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/08%3A_Laplace_Transforms/8.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/07%3A_Laplace_Transforms/7.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/6%3A_Power_Series_and_Laplace_Transforms/6.8%3A_Step_FunctionsIn this discussion, we will investigate piecewise defined functions and their Laplace Transforms. We start with the fundamental piecewise defined function, the Heaviside function.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/09%3A_Laplace_Transforms/9.04%3A_The_Shifting_Theorems\[\begin{aligned} {\cal L}^{-1}\left({3s+8\over(s+1)^2+4}\right)&= {\cal L}^{-1}\left({3s+3\over(s+1)^2+4}\right)+ {\cal L}^{-1}\left({5\over(s+1)^2+4}\right)\\&= 3{\cal L}^{-1}\left({s+1\over(s+1)^2+...\[\begin{aligned} {\cal L}^{-1}\left({3s+8\over(s+1)^2+4}\right)&= {\cal L}^{-1}\left({3s+3\over(s+1)^2+4}\right)+ {\cal L}^{-1}\left({5\over(s+1)^2+4}\right)\\&= 3{\cal L}^{-1}\left({s+1\over(s+1)^2+4}\right)+ {5\over2}{\cal L}^{-1}\left({2\over(s+1)^2+4}\right)\\&= e^{-t}(3\cos 2t+{5\over2}\sin 2t).\end{aligned}\nonumber\] \[Y(s)={1\over s[(s+2)^2+2]}+{1\over (s+1)[(s+2)^2+2]}={1/6\over s}+{1/3\over s+1}-{1\over 2}{s+2\over (s+2)^2+2}-{2\over 3}{1\over (s+2)^2+2}.\nonumber\]
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/13%3A_Laplace_Transforms/13.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/07%3A_Laplace_Transforms/7.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/07%3A_Laplace_Transforms/7.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/08%3A_Laplace_Transforms/8.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
- https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/03%3A_Topics_in_Differential_Calculus/3.03%3A_ContinuityIn the above figure, \(f\) is not continuous at \(x = x_2\) because \(\lim_{x \to x_2} f(x) \ne f(x_2)\); \(f\) is not continuous at \(x = x_3\) because \(\lim_{x \to x_3} f(x)\) does not exist (the r...In the above figure, \(f\) is not continuous at \(x = x_2\) because \(\lim_{x \to x_2} f(x) \ne f(x_2)\); \(f\) is not continuous at \(x = x_3\) because \(\lim_{x \to x_3} f(x)\) does not exist (the right and left limits do not agree—\(f\) is said to have a jump discontinuity at \(x = x_3\)); and \(f\) is not continuous at \(x = x_4\) because \(f(x_4)\) is not defined.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/06%3A_Laplace_Transforms/6.04%3A_The_Unit_Step_FunctionIn this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplac...In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.
