# Showing the function is Entire

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.

• Do you know Morera's and Fubini's theorems? – mrf Jan 15 at 13:55
• I know morera's theorem – MathCosmo Jan 15 at 13:58
• Essentially a duplicate of math.stackexchange.com/q/81949/42969. – Martin R Jan 15 at 14:00

$$\cos(tz)= \sum_{n=0}^{\infty}(-1)^n\frac{(tz)^{2n}}{(2n)!}$$
to see $$f(z)$$ is a power series that converges everywhere.