I have some questions on entire functions, now I know what entire functions are, however I have little clue or perhaps lack the know how on how to show such functions are entire, for instance:
Problem: $$\int_0^1 \frac{\sin zt}{t}\, dt $$
How can I show that these kind of functions are entire functions, when I get problems like these? I would appreciate the help.
Since\begin{align}\int_0^1\frac{\sin zt}t\,\mathrm dt&=\int_0^1\sum_{n=0}^\infty(-1)^n\frac{(zt)^{2n+1}}{t(2n+1)!}\,\mathrm dt\\&=\sum_{n=0}^\infty\frac{(-1)^nz^{2n+1}}{(2n+1)!}\int_0^1t^{2n}\,\mathrm dt\\&=\sum_{n=0}^\infty\frac{(-1)^nz^{2n+1}}{(2n+1)\times((2n+1)!)},\end{align}your function is entire.