IMEXP in Excel: Separate Complex Exponential Growth from Rotation

• Introduced: the first introduction version is not established by the reviewed official history.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official IMEXP reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMEXP(inumber)

Inumber is required and uses complex i or j notation. Microsoft recommends COMPLEX when the two input coefficients are already numeric.

The reference’s equation image gives EXP(x) times the quantity COS(y)+i SIN(y). Thus the real output coefficient is EXP(x)×COS(y), and the imaginary coefficient is EXP(x)×SIN(y).

Those trigonometric inputs are radians. If a source supplies degrees, convert that angle before constructing the complex exponent. Do not convert x, the real growth coefficient, as though it were another angle.

Enter only the first two numeric columns in A2:B5. Put =IMEXP(COMPLEX(A2,B2)) in C2 and fill down. The last two columns below are explanatory expected coefficients, rounded to nine decimals.

Extract the calculated coefficients into D and E with =IMREAL(C2) and =IMAGINARY(C2), filling down. Keep the underlying values; the table is a shortened display.

The third row reproduces Microsoft’s documented 1+i control at lower display precision. The fourth has the same real exponent and reversed angle, so its imaginary output changes sign while its real output stays the same.

In F2 use =EXP(A2)*COS(B2) and in G2 use =EXP(A2)*SIN(B2), filling down. These formulas independently reconstruct the two expected output coefficients from the documented equation.

For the first row, x is zero, so EXP(x)=1. Its output lies on the unit circle. For each of the remaining three rows, x is one and the output magnitude is e, approximately 2.718281828.

Check magnitude using =IMABS(C2). A nonzero imaginary input is not an instruction to take EXP of that coefficient as an additional growth factor. The angle and growth roles are separate.

IMREAL, IMAGINARY and IMABS let the result be audited without depending on its complete text rendering.

Suppose a fictional vector should have magnitude two and angle 60 degrees. Enter magnitude two in J2 and degrees 60 in K2. Use:

=IMEXP(COMPLEX(LN(J2),RADIANS(K2)))

The independently expected quantity has real coefficient one and imaginary coefficient the square root of three, approximately 1.732050808. Its magnitude is two.

The real exponent is LN(2), because EXP(LN(2))=2. The imaginary exponent is PI()/3, the radian equivalent of 60 degrees. Entering two directly as the real exponent would produce magnitude EXP(2), approximately 7.389056099, rather than magnitude two.

This construction requires a positive requested magnitude for LN. A zero magnitude cannot be obtained from a finite real exponent in the mathematical exponential model.

Let z be 0.5+0.25i. Its exponential has expected coefficients approximately 1.597466519 and 0.407900170. Squaring that result gives the exponential of 1+0.5i.

Compare:

=IMPRODUCT(IMEXP("0.5+0.25i"),IMEXP("0.5+0.25i"))
=IMEXP(IMSUM("0.5+0.25i","0.5+0.25i"))

Both have independently expected real coefficient approximately 2.385516731 and imaginary coefficient approximately 1.303213730. This is the exponential addition identity, checked through two different formula constructions.

Use a component tolerance, such as 0.000000001 for these moderate examples, instead of requiring exact equality of rounded complex strings. Multiplication and transcendental evaluation can introduce different final digits.

Adding 2×PI() to the imaginary exponent describes one complete rotation. The mathematical result is unchanged, so an exponent’s imaginary coefficient cannot be uniquely recovered from the exponential alone.

For example, exponents with imaginary parts zero and 2×PI(), both with real part zero, each model the quantity one. Native trigonometric rounding may leave a small residual imaginary coefficient; it should not be presented as exact character identity.

The dedicated page does not give a complete invalid-string, coercion or overflow table. Large positive real exponents can exceed representable magnitudes, while very negative ones can underflow; no native thresholds or error codes are certified here. No native Excel execution was performed.

• Microsoft IMEXP reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.