According to Euler's formula : we may write : So, Since is proven by Gelfond–Schneider theorem to be transcendental number it follows that is a transcendental number. So,my question is : Are numbers : transcendental numbers ?
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Originally Posted by princeps According to Euler's formula : we may write : So, Since is proven by Gelfond–Schneider theorem to be transcendental number it follows that is a transcendental number. So,my question is : Are numbers : transcendental numbers ? Yes, No, No. CB
Originally Posted by CaptainBlack Yes, No, No. CB Excuse me my ignorance but how can you conclude that just looking at numbers ?
Originally Posted by princeps Excuse me my ignorance but how can you conclude that just looking at numbers ? I don't conclude that from "just" looking at them. You have already demonstrated how you know the first is transcendental, the same form of argument can be applied to the others. CB
Originally Posted by CaptainBlack I don't conclude that from "just" looking at them. You have already demonstrated how you know the first is transcendental, the same form of argument can be applied to the others. CB But complex logarithm is multibranched .This means that there are many different possibilities for , specifically for any integer . In particular, is transcendental if you take any branch, except the principal branch, in which case it is an integer. Am I correct ?
Originally Posted by princeps But complex logarithm is multibranched .This means that there are many different possibilities for , specifically for any integer . In particular, is transcendental if you take any branch, except the principal branch, in which case it is an integer. Am I correct ? No logarithms (explicitly) involved: for Transcendental otherwise. CB
Last edited by CaptainBlack; Nov 30th 2011 at 11:56 PM.
Originally Posted by CaptainBlack No logarithms (explicitly) involved: CB But : If then and is proven transcendental number so number should be transcendental also . Am I missing something ?
Originally Posted by princeps But : If then and is proven transcendental number so number should be transcendental also . Am I missing something ? Opp.. principle values are not transcendental ... Now I think about it since they are all real I'm not sure how one would characterise the principle value in this case. CB
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