Research Problems in Function Theory — Problem 2.61

Definitions and mathematical statement

Let $\Gamma$ be a rectifiable Jordan curve in $\mathbb C$, oriented once around its bounded interior. A Jordan curve is a simple closed curve; rectifiability means finite arclength. For $a\in\mathbb C$ and $\theta\in\mathbb R$ let $\sigma_{a,\theta}(z)=a+e^{i\theta}z$ be an orientation-preserving Euclidean rigid motion, transporting the curve's orientation. Say that $\Gamma$ has the stated Morera property if every continuous complex-valued function $f$ on $\mathbb C$ satisfying \[ \int_{\sigma_{a,\theta}(\Gamma)} f(z)\,dz=0 \qquad\text{for every }a,\theta \] is entire. The integral is the complex line integral with differential $dz$, not an arclength integral $|dz|$ or an area integral. The conjecture asks whether circles are the only rectifiable Jordan curves lacking this property. A more restricted version asks the same among boundaries of bounded convex planar sets. Thus one seeks, for every eligible noncircular curve, an implication from the vanishing of all its congruent curve integrals to holomorphy of $f$. The source gives circles as counterexamples and several affirmative noncircular families. The curve is fixed: dilations are not included among the test motions.

Short English statement

Are circles the only closed simple rectifiable curves whose translated and rotated Morera tests fail to characterize entire functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302061 (AMR-022-2061), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.61. Original source indexed by AMR-022-2061. https://arxiv.org/abs/1809.07200.