How does the transistors amplifying function differ from when AC signal applied and just a DC voltage applied? Thanks in advance
There is a difference in how AC gain is expressed vs. DC gain. DC gain testing is done at a few micro-amps to a milliamp and is sometimes called the beta or hFE of a bjt transistor.
AC gain is expressed as hfe and is the current gain around a common point or device. AC gain is often taken at several major frequencies such as unity gain, gain bandwidth product and full power gain. Full power gain is just a few percent of unity gain, so often there are many charts in a transistors datasheet.
The designers want to make it clear where a given transistor is useful with gain vs. frequency vs. 1/f noise, etc. This answer here helps out with the hFE vs. hfe issue.