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.

Hence: .

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I think using the power series expansion for exp(i*x) would be more useful. This reduces to two infinite sums, one for cos(x) and one for i*sin(x). If I’m not mistaken, and I well could be, what you seem to have done is demonstrate that d(exp(x))/dx=exp(x) using Euler’s formula.

Thanks. I will have a fuller look, but your comment has made me notice I forgot to show why . I will edit the article now to get this right.

Actually the proof relies on but is not simply a restatement of it (the product rule is also needed). I think the concept which underlie this proof are more likely to be understood than Taylor series (ie I don’t know too much about Taylor series).

Thanks for your comments