Editing Bipolar coordinates (section)

== Proof that coordinate system is orthogonal ==

The equations for ''x'' and ''y'' can be combined to give

:<math>
x + i y = a i \cot\left( \frac{\sigma + i \tau}{2}\right)
</math><ref name="Polyanin"/><ref name="Happel"/>
or
:<math>
x + i y = a \coth\left( \frac{\tau-i\sigma}{2}\right).
</math>

This equation shows that ''σ'' and ''τ'' are the real and imaginary parts of an [[analytic function]] of ''x+iy'' (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory of [[conformal mapping]]) (the [[Cauchy-Riemann equations]]) that these particular curves of ''σ'' and ''τ'' intersect at right angles, i.e., it is an [[orthogonal coordinate system]].