Through direct AC computer system on immediately but through power button it doesn't on the computer system?

ok check the wire of power botton link through board

there is problem with power button.

power button error, try changing power button wire with reboot button wire, so reboot button will be power button then check your computer if it works then power button needed to be replace. 

Your question is not really clear , but if i got it right and this is what you wanted to say : is that when you connect the computer to the power it automatically turns on , but if you press the power button nothing happens (the computer doesn't turn on) Then here is my answer : As for the "Automatic power on" thing ,, i guess that's because of either that your mother board battery is empty or the cmos settings has been reset and not set again As for the "Power button" thing , i think if the power button doesn't power on neither power off the computer then probably the power button is either not functioning probably (corrupted) or not connected to the Mother Board I hope that helped answer your question .
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good luck