Which is, of course, the product rule.
Saw this referred to in A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity and it made me shudder – as I always seemed to struggle with it at ‘A’ level maths – so here, for my own benefit, is a quick proof/explanation of the method.
The general rule is:
And why is this so?
From the product rule we know:
And, integrate both sides and we have:
Of course, we still have to apply this sensibly to make our problem easier to integrate!
Reading An Introduction to Laplace Transforms and Fourier Series I reach the point where it is stated, rather axiomatically, that: .
This is a beautiful formula and has always suggested to me some sort of mystical inner mathematical harmony (yes, I am a materialist, but I cannot help it).
But these days I also want to see the proof, so here is one:
We know that complex numbers can be described in polar co-ordinates:
So too where and depend on .
Now (and applying the product rule)
So we equate the real and imaginary sides of both sides of this equality we have:
Then, recalling , we have
By inspection we can see that and , giving us:
and multiplying both sides by we have:
Reversing the signs:
But what of and ? Well, we have and .
So is constant with respect to while varies as .
If we set then – a wholly real number, so and . Thus and we can replace with throughout.
I know this is a piece of elementary calculus but I just worked it out from (more or less) first principles (as I knew what the answer was but did not know why).
Let what is ?
where is some constant. Hence or
and so and so .
And , so
Applying the chain rule and so
Now I have written it all out I know there are redundant steps in there, but this is how I (re)discovered it…
Not sure, so maybe someone who knows can tell me.
Following on from the last blog, can we show ?
Assume a constant, exists such that , could give us this ?
Edit: Professor Rubin (see comments) tells me that, as I feared, what follows is not supportable:
Now, and this is the bit I have most doubts over, as and then and and assuming then we are left with .
Hence and therefore , the constant we are seeking.
But there are a lot of assumptions in there: anybody able to tell me how valid they are?
Using the chain rule:
And , Euler’s number, really is magical.