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Complex Exponentiation


A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies

 (a+bi)^(c+di)=(a^2+b^2)^((c+id)/2)e^(i(c+id)arg(a+ib)),
(1)

where arg(z) is the complex argument. Written explicitly in terms of real and imaginary parts,

 (a+bi)^(c+di)=(a^2+b^2)^(c/2)e^(-darg(a+ib))×{cos[carg(a+ib)+1/2dln(a^2+b^2)]+isin[carg(a+ib)+1/2dln(a^2+b^2)]}.
(2)

An explicit example of complex exponentiation is given by

 (1+i)^(1+i)=sqrt(2)e^(-pi/4)[cos(1/4pi+1/2ln2)+isin(1/4pi+1/2ln2)].
(3)

A complex number taken to a complex number can be real. In fact, the famous example

 i^i=e^(-pi/2)
(4)

shows that the power of the purely imaginary i to itself is real.

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In fact, there is a family of values k such that (ik)^(ik) is real, as can be seen by writing

 (ik)^(ik)=e^(-kpi/2)[cos(klnk)+isin(klnk)].
(5)

This will be real when sin(klnk)=0, i.e., for

 klnk=npi
(6)

for n an integer. For positive n, this gives roots k_n or

 k_n=e^(W(npi)),
(7)

where W(z) is the Lambert W-function. For n>1, this simplifies to

 k_n=(npi)/(W(npi)).
(8)

For n=1, 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928), 4.30453, 5.51798, 6.63865, 7.6969, ....


See also

Complex Addition, Complex Division, Complex Multiplication, Complex Number, Complex Subtraction, Exponent, Exponent Laws, Exponential Function, Power

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References

Sloane, N. J. A. Sequence A088928 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Complex Exponentiation

Cite this as:

Weisstein, Eric W. "Complex Exponentiation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexExponentiation.html

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