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Complex Exponentiation

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Complex exponentiation raises a nonzero complex number to a complex number . Its principal value is , where uses the principal branch of the natural logarithm and . For real , , , and with , this gives

(1)

where is the complex argument. Written explicitly in terms of real parts 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. For example,

(4)

shows that the principal value of the imaginary unit raised to itself is real.

ComplexExponentiation

ComplexExponentiationReIm

ComplexExponentiationContours

For , the principal value of can be real, as can be seen by writing

(5)

This will be real when , i.e., for

(6)

for an integer. Since for , only is possible. These values give

(7)

where is the principal branch of the Lambert W-function. For , this simplifies to

(8)

For , 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, Principal Branch

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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 Resource. https://mathworld.wolfram.com/ComplexExponentiation.html

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