I'm looking for the article
Carleman, T. Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes. (French) Ark. Mat., Astr. Fys. 26, (1939). no. 17, 9 pp.
As it is very old I could not find it.
I wonder if anyone has a scanned copy of it.
You can find a copy in his "Edition compléte des articles", published by the Mittag-Leffler Institute: the book is still in print and if you have a few bucks, I advice you to purchase a copy of it, since Carleman was a true master in analysis. Also, a deeper search of World cat shows that there are many libraries around the world which hold copies of the journal: you can check which of them offer the document delivery service (which is usually free or almost free of charges), give the chosen library officers the paper complete reference and ask for a copy from the Journal. If you do not succeed following my previous advices, contact me directly by e-mail.