Engineering
Complex Polar Powers, Square Roots, Exponentials, and Natural Logs (IMPOWER, IMSQRT, IMEXP & IMLN)
Compute numeric real and imaginary parts of z^n, sqrt(z), exp(z), and ln(z) using De Moivre's theorem and Euler's formula with ATAN2, COS, SIN, EXP, and LN.
Why numeric (Re, Im) columns beat text-encoded IM* strings
Excel's engineering complex functions (IMPOWER, IMSQRT, IMEXP, IMLN, IMCOS, IMSIN) return formatted text strings like "-7+24i" that cannot be plotted, summed, or multiplied with standard spreadsheet operators.
Storing real part x (B2) and imaginary part y (B3) as numbers unlocks polar form r = SQRT(x^2 + y^2) and theta = ATAN2(x, y): De Moivre's theorem evaluates z^n = r^n * (cos(n*theta) + i*sin(n*theta)) for any real power n (including n = 0.5 for IMSQRT), IMLN evaluates ln(r) + i*theta, and Euler's identity evaluates exp(x + i*y) = exp(x)*(cos(y) + i*sin(y)).
| Operation on z = 3 + 4i (r = 5) | Real Component Formula | Real Result | Imaginary Result |
|---|---|---|---|
| z^2 (IMPOWER, n = 2) | r^2 * cos(2 * theta) | -7 | 24 |
| sqrt(z) (IMSQRT, n = 0.5) | r^0.5 * cos(0.5 * theta) | 2 | 1 |
| z^3 (IMPOWER, n = 3) | r^3 * cos(3 * theta) | -117 | 44 |
| ln(z) (IMLN) | LN(SQRT(x^2 + y^2)) | 1.6094 | 0.9273 rad |
De Moivre Real & Imaginary Parts of z^n (IMPOWER & IMSQRT)
Computes the exact real part of (B2 + i*B3)^B4 using modulus r = SQRT(B2^2 + B3^2) and phase angle ATAN2(B2, B3); replace COS with SIN for the imaginary part.
=LET(r_mod, SQRT(B2 * B2 + B3 * B3), th_ang, ATAN2(B2, B3), IF(r_mod = 0, 0, ROUND(POWER(r_mod, B4) * COS(B4 * th_ang), 4)))Principal Complex Natural Logarithm Real Part (IMLN)
Computes Re(ln(x + i*y)) = LN(SQRT(x^2 + y^2)), while ATAN2(x, y) gives Im(ln(x + i*y)) in (-pi, pi].
=ROUND(LN(SQRT(B2 * B2 + B3 * B3)), 4)Try the example
Use the original values or change them to test your understanding.