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Proof

(related to Proposition: Complex Conjugate of Complex Exponential Function)

• By hypothesis, $z\in\mathbb C.$ It follows:
• $\exp(z)^*=\left(\sum_{n=0}^\infty \frac{z^n}{n! }\right)^*$ by definition of the complex exponential function and the complex conjugate.
• $=\sum_{n=0}^\infty \left(\frac{z^n}{n! }\right)^*,$ by rules of calculating with complex conjugates.
• $=\sum_{n=0}^\infty \frac{(z^*)^n}{n! },$
• $=\exp(z^\ast),$ by definition of the complex exponential function and the complex conjugate.
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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983