Function Equal to Its Derivative

An alternative representation of the last integral can be expressed in plane polar coordinates r, 0. The two coordinate systems are related by 

The element of area in polar coordinates is given by rdrdO, so that the double integral becomes 

Integration over 9 gives a factor In. The integral over r can be done after the substitution u = j, du = Irdr  

Therefore, = 2n x = it, and Laplace s result (1.22) follows from the square root. 

The problem is to find a function /(x) that is equal to its own derivative / (x). Assume that /(x) can be expressed as a power series 

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