Showing the function is Entire

Question

Suppose $f$ be a continuous function on $[0,1]$. Define $g(z)= \int_0^1 f(t)cos(tz) dt$. Prove that $g(z)$ is an entire function. I'm a beginner in complex analysis and read upto cauchy integral formula. Please help. Thanks in advance.

Answer 1

Hint: Use

$$\cos(tz)= \sum_{n=0}^{\infty}(-1)^n\frac{(tz)^{2n}}{(2n)!}$$

to see $f(z)$ is a power series that converges everywhere.