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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.

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

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