- #1
rsq_a
- 107
- 1
Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented.
In examining the function
[tex]
f(x) = \frac{1}{1 + x^2}
[/tex]
we can derive the series expansion
[tex]
\sum_{n=0}^\infty (-1)^n x^{2n}
[/tex]
We note that the ratio test (which does not involve complex numbers) indicates that the series necessarily diverges if
[tex]
| -x^2 | > 1
[/tex]
or [itex]x > 1[/itex].
However, returning to the function f(x), we see that the point x = 1 is not at all special. How is the specialness of x = 1 explained without the notion of a complex number?
Perhaps we are misleading the reader by our choice of a series expansion about x = 0. Had we done a series expansion about x = 1, then we would have found that there is something special about the points [itex]x = 1 \pm \sqrt{2}[/itex]. But the question remains, how can this be anticipated a priori, without complex numbers?
* Note that in the language of complex numbers, this is explained simply by the fact that the distance to the nearest singularity, x = i, is 1. But this involves the notion of the Argand plane.
In examining the function
[tex]
f(x) = \frac{1}{1 + x^2}
[/tex]
we can derive the series expansion
[tex]
\sum_{n=0}^\infty (-1)^n x^{2n}
[/tex]
We note that the ratio test (which does not involve complex numbers) indicates that the series necessarily diverges if
[tex]
| -x^2 | > 1
[/tex]
or [itex]x > 1[/itex].
However, returning to the function f(x), we see that the point x = 1 is not at all special. How is the specialness of x = 1 explained without the notion of a complex number?
Perhaps we are misleading the reader by our choice of a series expansion about x = 0. Had we done a series expansion about x = 1, then we would have found that there is something special about the points [itex]x = 1 \pm \sqrt{2}[/itex]. But the question remains, how can this be anticipated a priori, without complex numbers?
* Note that in the language of complex numbers, this is explained simply by the fact that the distance to the nearest singularity, x = i, is 1. But this involves the notion of the Argand plane.
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