This may give you what you want.
expr = (2^(-3 + 4/m) E^(-((3 I \[Pi])/m)) (-1 +
E^((2 I \[Pi])/m))^3 Gamma[(-2 + m)/m] Gamma[(-1 + m)/
1/m + (I w3)/
2] (Beta[(-1 + m)/m, 1/(2 m) + 1/4 I (w1 + w3)] Gamma[
1/4 (4 - 2/m - 2 I w1 + I w3)] Gamma[(2 + 2 I m w1 -
I m w3)/(4 m)] -
Beta[(-1 + m)/m,...
Maybe the following? I didn't understand all the extra variables, so I stripped down the code more. In a complicated case, you might need Dynamic@Refresh instead of just Dynamic; see What is the point of Refresh if Dynamic has an UpdateInterval option? and related Q&A on Refresh. You'd probably have to put the output in a docked cell, since the FE ...