Manuals/calci/IMPOWER

IMPOWER(z,n)

• $z$ is the complex number is of the form $x+iy$
• $n$ is the power value.

Description

• This function gives the value of powers of complex number.
• DeMoivre's Theorem is a generalized formula to compute powers of a complex number in it's polar form.
• $i$ is the imaginary unit, $i={\sqrt {-1}}$
• Then the power of a complex number is defined by

$(z)^{n}=(x+iy)^{n}=r^{n}*e^{in\theta }=r^{n}(cosn\theta +isinn\theta )$

where $r={\sqrt {x^{2}+y^{2}}}$ and $\theta =tan^{-}1(y/x)$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta∈(-\pi,\pi]} .

• This formula is called DeMoivre's theorem of complex numbers.
• We can use COMPLEX function to convert real and imaginary number in to a complex number.
• In IMPOWER(z,n), $n$ can be integer, fractional or negative.
• If $n$ is non-numeric, function will return error value.

Examples

1. =IMPOWER("4+5i",3) = -235.99999+115i
2. =IMPOWER("9-7i",4) = -14852-8063.999999i
3. =IMPOWER("6",9) = 10077696
4. =IMPOWER("i",10) = -1+6.1257422745431E-16i

See Also

• IMREAL
• IMSUM
• IMAGINARY
• COMPLEX

References

De Moivre's formula