https://mathweb.ucsd.edu/~jmckerna/Teaching/19-20/Spring/120B/l_22.pdf

The Schwartz Reflection Principle, as formulated by G. Valiron in his book "Fonction Analytiques" (1954), is a fundamental result in complex analysis. The principle states that if a function \( f \) is holomorphic in the upper half-plane \( \text{Im}(z) > 0 \), continuous on the real axis \( \text{Im}(z) = 0 \), and satisfies certain growth conditions, then it can be extended to the lower half-plane \( \text{Im}(z) < 0 \) and this extension will be real on the real axis.

To find a proof of this principle, you might want to consult textbooks on complex analysis or research papers that discuss this topic. Some classic references for complex analysis that cover the Schwartz Reflection Principle include:

1. "Complex Analysis" by Lars Ahlfors

2. "Functions of One Complex Variable" by John B. Conway

3. "Complex Variables and Applications" by James Ward Brown and Ruel V. Churchill

These texts often provide detailed proofs and discussions of important theorems in complex analysis, including the Schwartz Reflection Principle.

Additionally, you may find papers or lecture notes available online that discuss the proof of the Schwartz Reflection Principle in detail. Searching academic databases such as Google Scholar or MathSciNet using relevant keywords like "Schwartz Reflection Principle proof" or "analytic continuation complex analysis" could yield helpful resources.

If you're unable to find a specific proof in these resources, you might consider reaching out to experts in the field of complex analysis for guidance or references to relevant literature.

Thanks for the replay! The essence of the question is that the following is not necessary to be supposed and still the assertion holds:

"continuous on the real axis \( \text{Im}(z) = 0 \), and satisfies certain growth conditions"

it seems this is a folklore for which no reference is easy to be found in public places.