It is possible to transform this equation to a form that is technically nonsingular see Kythe [ 10 ], for example , but the removable singularity that results involves the computation of values as the ratio of small quantities, even if one manages to avoid. We give a second example that requires more ingenuity; the difficulties will mean that Mathematica takes a little longer, but it can still cope. This is also the exact solution if, and only if, the domain in the -plane is a circular disk centered on the origin. As an example of an integral equation that is both nonlinear and singular, we consider the one introduced by Theodorsen for this purpose. This Demonstration plots the image of the unit disk under various conformal maps , normalized so that and. I think this must be a bug.
Make your selections below, then copy and paste the code below into your HTML source. This seems to work for The problem reduces to finding the dependence between these two angles; once we have determined the function , the mapping itself can easily be constructed. A standard result of complex analysis states that every injective analytic function of a complex variable is a conformal mapping onto its image, and conversely that every conformal mapping is an analytic function of a complex variable. I’ve noticed that when I remove the constant term in the denominator the problem goes away. Save to My Widgets. And here we obtain the maximum absolute value of the error at a random distribution of interior points. We appreciate your interest in Wolfram Alpha and will be in touch soon.
We appreciate your interest in Wolfram Alpha and will be in touch soon. Many of the functions here can be derived using Schwarz—Christoffel integrals. At points inthe mapping function is ;lot in mathemtica of via the Cauchy integral formula, while on the relationship between and is more immediate.
In fact, because of the extra difficulties we now have at the ends, we will push points even closer to them by using new[new[ ]].
Here is a parameter satisfying. Because of the symmetry, we can concentrate on the interval ; we know that is equal to plus an odd function of that must vanish at both and. We see that we have nonnegligible distortion, for the image domain is not convex.
We can also obtain an estimate of the overall accuracy in the interior by finding the mathematoca absolute value of the error at a random distribution of interior points.
Conformal Mapping of the Unit Disk
As far as I can tell, this artifact is shaped as if it is a white region between two kissing large-radius circles. Unaided, PolarMap does not deal with this example very effectively; it cuts off the corners of the square.
Because of the maximum modulus principle we, of course, expect the maximum error to occur on the boundary. Each plot shows the images of concentric circles around the origin and radial lines emanating from the origin. To add a widget to a MediaWiki site, the wiki must have the Widgets Extension installed, as well as the code for the Wolfram Alpha widget. The map we seek takes points on the boundary of the disk in the -plane identified by the angle onto points on the boundary of the domain in the -plane identified by the angle.
And here we obtain the maximum absolute value of the error at a random distribution of interior points. Neither do I get an artifact if I just trace the individual grid lines. To try to obtain a uniform distribution of errors we should, as before, concentrate the interpolation points near the ends of the interval, and therefore introduce a nonlinear scaling similar to one used before.
Howell, “Dictionary of Confomal Mapping. We can now plot the absolute value of the error on the boundary to assess the accuracy of our approximation there.
Make your selections below, then copy and paste the code below into your HTML source. The famous Riemann mapping theorem states that any simply connected domain in the complex plane other than the whole plane itself is conformally equivalent to the unit disk.
I think this must be a bug.
To add the widget to Blogger, click here and follow the easy directions provided by Blogger. However, finding an explicit conformal map for a given domain can be a tedious task. You will then see the widget on your iGoogle account.
The Mathematica Journal: Volume 9, Issue 2: Integral Equations
We can test the error at various points as in the previous example. The study of conformal mappings on the unit disk led to a classical branch of mathematics called univalent functionswhich was pursued for most of the twentieth century. For the domain is a circular disk of unit radius, while as the domain degenerates to two circular discs of radius touching tangentially. We give a second example that requires more ingenuity; the difficulties will mean that Mathematica takes a little longer, but it can still cope.
This is much as before, but Mathematica now appreciates being warned to expect trouble jathematica NIntegrate at points corresponding to the corners. The integral here conformxl to be interpreted as a Cauchy principal value, there being a singularity at. It is natural to approximate this function using a linear sum of functions of the formwhere must be an even integer.