Hello, my name is Daniel Friedrich. And in this short video, I'm going to show you how to calculate the Laplace transform of a few simple functions. So let's start with 'Q1'. Our task is to calculate the Laplace transform of the ramp, which is given by 'f(t)' is equal to 'α*t', where α is a real constant. So we can do this, directly from the Laplace transform definition. So we take the Laplace transform of 'α*t', and we plug it into the definition of the Laplace transform. So we have the integral from 0 to infinity, 'α*t times 'e^-(st) dt'. To solve this, we have to use integration by parts. So we write 'u(t)' is equal to 't', 'du' equal to one. 'dv' we use the 'e^-(st)'. And this is follows that  is 'v' is '-e^-(st)' over 's'. So do this, we can get our simplified integral. So the first part, '((-α*t)/s)*e^-(st) from 0 to infinity. plus 0 to infinity of  '(α/s)*e^(-st)dt' In this step I have taken the minus, which is normally here, And the minus in this term to already cancel them out and make them plus. So now we need to handle the limit. So we replace the infinity in the upper limit. with capital 'T' goes to infinity. You just keep the first part. And we directly calculate the integral of this. So this will give us, '(-α/(s^2))*e^-(st)', 0 to capital 'T'.   So now we can insert the limits, into the term here in the square brackets. And we get, for the upper limits, we would get, '((-α*T)/s)*e^(-sT)', minus '(α/(s^2))*e^(-sT)', minus the one for the lower limit. Say if we put in 0 here, we have 'α*t' over times 0,  this term is 0. While this term here, we have the 'e' to the power of minus 's*0', which is one. So we are left with '-(-α/(s^2))' These terms, as we have seen previously. This goes to 0, as long as the  real part of 's' is larger than 0. This part also goes to 0 as long as the real part of 's' is larger than 0. With all this together, we get that our Laplace transform of the ramp is just 'α/(s^2)',  for real part of 's' larger than 0. And this is a solution for  question one for this little video. Let's have a look at the second part. So we want to calculate the Laplace  transform of the sine and cosine functions. Here, 'β' is again a real constant. So let's look at this by remembering Euler's formula. Which is 'e' to the power of '(j*β*t)', is equal to 'cos(β*t) + j*sin(β*t)'.  So from this, we can see that our original function, 'f' is just the real part, is just the imaginary part of this exponential. And cosine is the real part. The imaginary part of 'e^(j*β*t)', and the cosine is the real part, 'e^(j*β*t)'. So now we have this  and we already know what the Laplace transform of the exponential function is. So we can reuse this. How can we do this? So we set 'α' is equal to 'j*β'. And then we calculate  the Laplace transform of 'α*t'. Which, this is the Laplace transform of 'e^(j*β* t)'. And this is, '(1/(s-α))'. But we can now replace the 'α' again, with 'j*β'. So this,  '1/(s-(j*β))'. So now we have the Laplace transform of this. Now we just need to find the real  and imaginary part of this. The easiest way to do this is to multiply, with the complex conjugate of this part of the bottom here. So, we multiply this with 's + (j*β)'. So this term here is basically just one. So we didn't change our equation. But now we can get rid of the complex number, in our fraction. So this becomes,  '(s+(j*β))/(s^2+β^2)'. And this is for the real part of 's' larger than 0. So now we have the Laplace transform of our exponential of, 'j*β*t'. So now, we just can find the real and imaginary part to find the Laplace transforms of the sine and cosine. So our Laplace transform of our sine, 'β*t', is just the imaginary part. So it's just,  'β/(s^2+β^2)'. Again, real part of 's' larger than 0. And the Laplace transform of, 'cos(β*t)', is the real part. So it's,   's/(s^2+β^2)'. And in this way we have used the Laplace transform of the exponential, which we already knew to calculate the Laplace transform of sine and cosine functions. In the final example of this little video, we're looking to calculate the Laplace transform of the hyperbolic sine and hyperbolic cosine functions. So we have hyperbolic sine of 'β*t', hyperbolic cosine of 'β*t', with β being again a real number. So let's look at the hyperbolic sine first. So we can, we know we can write these hyperbolic functions, as a combination of exponentials. So we can write, 'f(t)', which is the hyperbolic sine of 'β*t'. This is equal to one half, times, '(e^(β*t)-e^(-β*t))' And again,we know the Laplace transform of these terms. We know the Laplace transform is linear so that we can use, these two pieces of knowledge to calculate the Laplace transform of the sine hyperbolic. So, we know Laplace transform of this is one. This for real part of 's', larger than 'β'. And we also know that this is one over 's' plus 'β', for the real part of 's' larger than minus 'β'. So we can combine these. So our Laplace transform of the hyperbolic sine 'β*t' is the Laplace transform of this part  minus the Laplace transform of this part. So we get, '0.5(1/(s-β)-1/(s+β))' So we can put these to bring them on the same fraction. So basically multiply this part with 's+β', on the numerator and denominator. This we multiply with 's-β' on the numerator and denominator. So this will give us, I'll leave the one half here. So we get, '0.5*((s+β)-(s-β))/((s-β)*(s+β)) And we can simplify this to get one half, So we have 's' minus 's',  so the 's' cancel out. So we have '2*β',  at the bottom we have 's^2 -  β^2'. And then the 2's cancel out as well. So we are left with,  'β/(s^2-β^2)'. And this is valid as long as the real part of 's' is larger than the absolute value of β, because we need to take the largest value of these two. So if β is positive, this will be this one  and if β is negative, it will be this term. So in this form, we have calculated the Laplace transform of the hyperbolic sine. So in the similar way, you can calculate the Laplace transform of the hyperbolic cosine. Hyperbolic cosine is given by, '0.5*(e^(β*t)+e^*(-β*t))'. So the only thing which changes is that there is a plus here instead of the minus for the hyperbolic sine. So you follow the same steps and you should arrive at the Laplace transform of the hyperbolic cosine 'β*t'. This should be 's' over 's^2' minus 'β^2', for the real part of 's' is larger than the absolute value of 'β'. So the steps of this, please follow these steps, follow the steps for the hyperbolic sine and then you should arrive at this. If you have any questions about this example or any other part of the course, please head to discussion forum where you can discuss it. With everyone else in the class, the tutors, and David and myself.  Thank you!