Explicit Roots of Transcendental Equations

Cauchy's integral theorem [mathworld.wolfram.com/CauchyIntegralTheorem.html] states that if is analytic in some simply connected region , then

for any closed contour completely contained in . The roots of a function can be determined by locating the singularities of the reciprocal of the function, . If has a simple pole at , then is analytic. Luck and Stevens [1] use this to obtain an explicit expression for the root of ,

where the contour contains the single pole at .

For example, consider computing the roots of . The roots lie on the real axis.

To determine the first root, we use equation (2), integrating around an arbitrary contour enclosing just this root.

We verify that this value is correct to machine precision.

Alternatively, evaluating both integrals around the same circular contour

equation (2) becomes

where . The values of and do not matter as long as the contour circumscribes the root.

To determine the second zero of , we choose a circle centered at with radius that circumscribes this root and no other.

We use equation (3) to determine .

This is an excellent approximation to the second root.

The th complex Fourier coefficient is defined by [mathworld.wolfram.com/FourierSeries.html]

so equation (3) reduces to

involving a simple ratio of Fourier coefficients.

To determine the third zero of , we choose a circle centered at with radius that circumscribes this root and no other, computing the Fourier coefficients approximately using Fourier, sampling uniformly over just 16 times.

We use equation (4) to determine .

Then we check that this value is a good approximation to the third root.

Increasing the number of sample points improves the accuracy of roots computed via Fourier.