The u-invariant of one-dimensional function fields over real power series fields

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The u-invariant of a field K of characteristic not 2 is the largest dimension of an anisotropic torsion quadratic form over K. We extend the recent theorem by Harbater, Hartmann, and Krashen on the u-invariant of a one-dimensional function field over a complete discretely valued field to the case where the residue field can be ordered.