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.
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