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Is there a conformal map from a sector in the circle with 60 degrees angle to a equilateral triange that maps the points 0, 1 and $exp (\frac{\pi i}{3})$ to the vertices of a triangle?

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    Yes, as long is the interiors are concerned. The [Riemann mapping theorem](http://en.wikipedia.org/wiki/Riemann_mapping_theorem) guarantees this.2012-10-13

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