Difference between revisions of "U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n"
From specialfunctionswiki
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==References== | ==References== | ||
| − | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n|next=T n(x)=Sum (-1)^ | + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n|next=T n(x)=Sum (-1)^kn!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)}}: Theorem 7.1 (ii) |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Revision as of 19:22, 15 March 2018
Theorem
The following formula holds: $$U_n(x) =-\dfrac{i}{2} \left[ \left( x + i \sqrt{1-x^2} \right)^n + \left( x-i\sqrt{1-x^2} \right)^n \right],$$ where $U_n$ denotes Chebyshev U and $i$ denotes the imaginary number.
Proof
References
- 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 7.1 (ii)
