03/02/2023, 07:21 PM
Nice work you have here-- this gives us a unique way to connect these series outside their natural boundary into a function inside the natural boundary!
Also, based on your work, we get the 'Cauchy-like' integral formula
\[ F(x) = a_0 + \int_C \frac{F(z)}{z-\frac{1}{x}}dz\]
Where the contour C is taken outside the natural boundary and the pole at \(z = \frac{1}{x}\) needs to outside the contour integration. Actually, we could do that, or we could only pick up the pole at \(z = \frac{1}{x}\) is pick up no other residue (but then the constant term is a bit more complciated I think). Either way, we now have an easy way to use contours only inside of the natural boundary to compute values on the inside!
Also, based on your work, we get the 'Cauchy-like' integral formula
\[ F(x) = a_0 + \int_C \frac{F(z)}{z-\frac{1}{x}}dz\]
Where the contour C is taken outside the natural boundary and the pole at \(z = \frac{1}{x}\) needs to outside the contour integration. Actually, we could do that, or we could only pick up the pole at \(z = \frac{1}{x}\) is pick up no other residue (but then the constant term is a bit more complciated I think). Either way, we now have an easy way to use contours only inside of the natural boundary to compute values on the inside!
