• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Taylor series for i^i?

  • Thread starter Khan
  • Start date

Khan

I'm having some problems expanding i^i, could anyone help? I know it becomes a real number somehow, and I'm familiar with the e^(i * pi) expansion, but is the i^i done in the same way?
 

Njorl

Science Advisor
245
10
There is a well known expansion for a^x:

a^x=SUM[((alnx)^n)/(n!)]

Just replace a and x with i.

At first glance, it doesn't look real to me, but maybe the sum telescopes.

Njorl
 

Soroban

Hello, Khan!

I'm not sure what you mean by expanding ii,
since it is already a constant.

Using DeMoivre's Theorem (Euler's?): eix = cos x + i sin x,
when x = pi/2, we have: ei*pi/2 = cos(pi/2) + i sin(pi/2) = i

Raise both sides to the power i: ii = (ei*pi/2)i= e-pi/2 = 0.207879576...
 

Related Threads for: Taylor series for i^i?

  • Posted
Replies
3
Views
1K
  • Posted
2 3
Replies
57
Views
9K
  • Posted
Replies
3
Views
1K
  • Posted
Replies
1
Views
774
  • Posted
Replies
2
Views
2K
  • Posted
Replies
1
Views
2K
Replies
3
Views
9K
Replies
2
Views
7K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top