# Laplace Transform of Some Basic Functions

*Exponential Function*

By definition of Laplace transform,

Similarly, for

*f*(*t*) =*e*^{â€“at},*Unit Step Function or, Heaviside Unit Function*

*f*(

*t*) =

*u*(

*t*) = 1 for

*t*> 0

= 0 for
and is undefined for

*t*< 0*t*= 0.**(a) Unit step function**

**(b)**

*Shifted unit step function*Also, the Laplace transform of step function of magnitude

*K*isSimilarly, the Laplace transform of the shifted unit step function

*u*(*t*â€“*T*) is, {by differentiation theorem}

Another function, called

*gate function*can be obtained from step function as follows.**Gate function**

Therefore,

*g*(*t*) and,*The Sine Function*

*f*(

*t*) =

*F*(

*s*) =

*The Cosine Function*

*f*(

*t*) =

*F*(

*s*) =

*The Hyperbolic Sine Function*

*f*(

*t*) =

*F*(

*s*) =

*The Hyperbolic Cosine Function*

*f*(

*t*) =

*F*(

*s*) =

*The Damped Sinusoidal Function*

*f*(

*t*) =

*F*(

*s*) =

=

=

*The Damped Cosine Function*

*f*(

*t*) =

*F*(

*s*) =

=

=

*The Ramp Function*

**Ramp function**

Integrating by parts, let,

*u*=

*t*and

^{n}*dv*=

*e*

^{â€“ st}

*dt*

then and

Now,

*F*(*s*) ==

=

=

For

*n*=1,For

*n*=2,*Impulse Function or Dirac Delta Function**[**Î´**(t)]*

**Generation of impulse function from gate function**

It is a function of a real variable

*t*, such that the function is zero everywhere except at the instant*t*= 0. Physically, it is a very sharp pulse of infinitesimally small width and very large magnitude, the area under the curve being unity.Consider a gate function as shown in figure.

The function is compressed along the time-axis and stretched along the

*y*-axis, keeping area under the pulse unity. As*a*â†’0, the value of and the resulting function is known as impulse.It is defined as,

*Î´**(**t*) = 0 for*t*â‰ 0and

Also,

*Î´**(**t*) =The Laplace transform of the impulse function is obtained as,

[

*Î´**(**t*)]= 1