Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin t}{t}J_m(t)\,dt$

Question

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin(t)}{t}J_m(t)\,dt$

$$\int_R \frac{\sin(t)}{t}J_m(t)\,dt=\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt =\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt = c_mx^m \int_{-1}^1 \hat{\chi_{{(-1,1)}}}(t)e^{its}(1-x^2)^{m-\frac{1}{2}} \,dt$$

How do I contunie from here?

Answer 1

For any integer $m>0$, the result is $0$. In case $m=0$ the result is $\pi$