The Laplace transform F(s) of the exponential function. f(t) = eat when t ≥ 0, where a is a constant and (s - a) > 0, is
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Transform Theory
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Evaluate
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If the Laplace transform of is the Laplace transform of tcosh t is
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The function f(t) satisfies the differential equation and the auxiliary conditions, f(0) = 0, The Laplace transform of f(t) is given by
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The Laplace transform of ei5t where is
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Laplace transform of the function sin ωt is
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For the function {\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}}
{ - 2,}&{ - \pi < {\text{x}} < 0} \\
{2,}&{0 < {\text{x}} < \pi }
\end{array}} \right.
The value of an in the Fourier series expansion of f(x) is
The value of an in the Fourier series expansion of f(x) is
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Laplace transform of cos (ωt) is The Laplace transform of e-2t cos(4t) is
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The Laplace transform of sinh(at) is
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The Fourier cosine series for an even function f(x) is given by
The value of the coefficient a2 for the function f(x) = cos2(x) in [0, π] is
The value of the coefficient a2 for the function f(x) = cos2(x) in [0, π] is
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A solution for the differential equation with initial condition x(0-) = 0 is
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Consider the differential equation with and
The numerical value of is
The numerical value of is
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A delayed unit step function is defined as {\text{u}}\left( {{\text{t}} - {\text{a}}} \right) = \left\{ {\begin{array}{*{20}{c}}
{0,}&{{\text{for t}} < {\text{a}}} \\
{1,}&{{\text{for t}} \geqslant {\text{a}}}
\end{array}} \right.. Its Laplace transform is
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The Laplace Transform of f(t) = e2t sin(5t) u(t) is
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Let be the Laplace Transform of a signal x(t). Then, x(0+) is
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If f(t) is a function defined for all t ≥ 0, its Laplace transform F(s) is defined as
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Laplace transform for the function f(x) = cosh(ax) is
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If F(s) is the Laplace transform of function f(t), then Laplace transform of is
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The inverse Laplace transform of is
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The Fourier series of the function,
\begin{array}{*{20}{c}} {{\text{f}}\left( {\text{x}} \right) = 0,}&{ - \pi < {\text{x}} \leqslant 0} \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \pi - {\text{x,}}}&{0 < {\text{x}} < \pi } \end{array} in the interval is
The convergence of the above Fourier series at x = 0 gives
\begin{array}{*{20}{c}} {{\text{f}}\left( {\text{x}} \right) = 0,}&{ - \pi < {\text{x}} \leqslant 0} \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \pi - {\text{x,}}}&{0 < {\text{x}} < \pi } \end{array} in the interval is
The convergence of the above Fourier series at x = 0 gives
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