Discussion turned into argument with the conference keynote speaker, would it be appropriate to email an apology? I had a chance to visit as an audience to an international conference in computer science. As usual, there were few keynotes before the scheduled paper presentation is carried out for the day.

During the presentation of one keynote speech, where the speaker (quite renowned in the field) made a very strong argument that the equation/method that he is using will solve a lot of problems. However, I could point that there was a mistake in the method (rather the equation) itself. The method did not obey the basic laws to solve any possible problems.

The discussion with the Professor continued for a while and he never agreed to what I was saying and I also never accepted those not-so-possible facts. This whole thing converted into an argument. 

I understand that it happens during such process because of adrenaline rush. Now, after few days of wondering about it, I am thinking that I made a mistake. I am feeling bad. Could it hamper my future/present career prospectives? I am presently in my PhD research. I want to mail to the professor and say sorry for the incident. Not sure.
Academia is one of a few areas where it is still OK to have a discussion and even an argument. In fact, some research groups are involved in long-standing discussion / heated competition / argument. Academic discussion is considered a way to improve our joint understanding of the world and to further the scientific progress and knowledge.

However, as in any game, there are some rules to understand and follow. 


It is OK to disagree with someone's argument, but you should listen and do your best to understand it first. If you don't understand something, ask politely. If you still don't understand, ask someone for a third opinion. If necessary, allow yourself time to think before you continue the discussion.
It is very important to be polite and not to become personal.
Seniority per se does not matter so much: professors do make mistakes sometimes, and they know it. However, it is important to remember that most of professors are called professors for a good reason, and they are indeed experts in their field. Their words are not always infallible, but usually quite well thought through. It is OK to ask a professor to explain his/her statement, but it is good to show that you generally respect and trust them as colleagues and experts.


As long as you followed these rules, I think you are fine and there should be no hard feelings. Otherwise, it could be a good idea to write back with a sort of apology. Professors are people first, and as all people, they have feelings, which should be respected. So if you fill you hurt their personal feelings, an apology would be nice.