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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Do you know Morera's and Fubini's theorems? – mrf Jan 15 '19 at 13:55
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I know morera's theorem – MathCosmo Jan 15 '19 at 13:58
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2Essentially a duplicate of https://math.stackexchange.com/q/81949/42969. – Martin R Jan 15 '19 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.
zhw.
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