Zoho Sheet IMEXP: Exponential of a Complex Number

• Introduced: The original release and date are not established by the checked official sources.
• Availability: Currently documented in Zoho Sheet’s rolling web-service help; there is no verified final supported-version endpoint.
• Scope: The English cloud reference does not establish identical behavior in every mobile client or imported workbook.
• Checked: October 14, 2026, against the official IMEXP reference. Documentation and independent arithmetic checks; no native Zoho Sheet test.

=IMEXP(complex_number)

All arguments are required; no optional parameter or default is documented in the official IMEXP reference.

Use a fresh worksheet for each function so the results in column C do not overwrite another exercise. Enter the following in A1:B10, or import the two-column practice CSV at A1. Keep column A as text and column B as numbers. The literal text does not include enclosing quotation marks in cells.

Rows 7, 8 and 10 are optional boundary inputs, not part of the worked result checks. In particular, do not fill IMLN through the zero row and assume it has a logarithm.

For real a and b, complex exponentiation has the mathematical identity

exp(a + bi) = exp(a) × [cos(b) + i sin(b)]

exp(a) controls magnitude (growth or decay); b controls phase in radians. Thus if a=0, the result lies on the unit circle in exact mathematics. Zoho’s documentation itself gives the same relationship when describing IMEXP, and it calls the single argument a text-form complex number.

Three published examples

These numeric output strings are from the Zoho example table. Independently evaluating e²·cos(3) and e²·sin(3) produces approximately −7.3151101 and +1.0427437, respectively. The second example’s positive imaginary component is correct because sin(−4) is positive, not negative. Do not infer a sign error from the negative angle without checking the trigonometry.

Use the practice inputs above. A2 contains 2+3j, A3 contains 1-4i, A4 contains 5j, A5 contains 3+4j and A6 contains 2-1j. Enter these formulas in cells C2:C6:

Your live workbook may serialize coefficients with different precision; check approximate numeric agreement rather than comparing exact long strings.

For A5 = 3+4j, Euler’s formula gives e³(cos 4 + j sin 4). Using e³≈20.08553692, cos 4≈−0.65364362 and sin 4≈−0.75680250 produces real −13.12878308 and imaginary −15.20078446. Its magnitude equals e³, not e^4 or the length of the input 3+4j.

For A6 = 2−j, the coefficient calculations are e²·cos(−1) and e²·sin(−1), approximately 3.99232405 and −6.21767631. The sign of the imaginary part follows the negative sine.

Signal phase representation

A pure imaginary exponent represents a point on the unit circle: exp(iθ)=cos θ+i sin θ. In a training workbook, put the text 5j in A4 without enclosing quote characters and calculate =IMEXP(A4). Compare its independently calculated magnitude to 1. If a phase angle comes from degrees, first convert it to radians mathematically; the exponential identity does not interpret b as degrees.

Growth and decay envelopes

For z=−0.5+2i, mathematical exp(z) has magnitude e^−0.5≈0.60653066 and phase 2 radians. A live formula proposal is =IMEXP("-0.5+2i"). Its independently expected coefficients are approximately -0.2524058153+0.5515167682i; output precision may differ. Exponential growth or decay is an interpretation of the real part, not proof that a measured business or physical process actually follows the mathematical model.

Round-trip with the complex natural logarithm

For nonzero z, exp(Log(z))=z in the usual principal-log convention; the inverse direction Log(exp(z)) may differ from z by an integer multiple of 2πi because angle is periodic. To evaluate the first identity in Zoho, enter =IMEXP(IMLN(A2)) in E2; its mathematical expected value is 2+3j. Both IMEXP and IMLN are officially documented; compare coefficients within a numerical tolerance rather than exact strings. Never write that IMLN(IMEXP(z)) invariably reproduces the original input’s imaginary part.

Magnitude verification

Mathematically, |exp(a+bi)|=exp(a). With A2=2+3j, the expected magnitude is e². Enter =IMABS(IMEXP(A2)) and =EXP(2) in separate empty cells; both have mathematical expectation 7.3890560989. The related IMABS and EXP articles establish the respective functions, but these comparisons check the intended magnitude relationship.

Data export and text parsing

Complex results are displayed in textual i/j notation in Zoho’s examples. If sending calculated outputs to CSV or another spreadsheet package, consider exporting real and imaginary coefficients separately after validating IMREAL and IMAGINARY. This avoids relying on a destination application’s parser for a text-form complex number. Do not assume that a result is an Excel-compatible numeric complex object.

• Angle units: The mathematical IMEXP identity uses radians. A phase value measured in degrees must be converted before the formula is evaluated; do not confuse 180 degrees with 180 radians.
• Very large positive real part: Exponential magnitude grows quickly. Zoho’s help does not specify a reliable overflow threshold, maximal exponent, or exact error for extreme arguments.
• Very negative real part: Magnitudes can become very small. Underflow and display rounding are application- and precision-dependent, with no threshold established in the official function page.
• Repeated exponentials: Nested IMEXP(IMEXP(z)) is mathematically defined but can rapidly change scale; it is not documented as a special array/vector operation.
• Input grammar: The function documents one text-form complex number and examples with lowercase i/j. Unsupported number-text conventions, uppercase units or embedded spaces require testing.
• Whole-range input: A cell reference with one complex value is a sensible formula proposal, but passing a range to a singular-argument IMEXP function is not established here.
• Output formatting: Tiny imaginary coefficients close to zero may display as numerical noise, an omitted component, or a longer decimal. No exact serialization rule has been tested.
• Complex power confusion: IMEXP(z) means e^z, whereas IMPOWER(z;n) means z raised to a supplied exponent. They are not interchangeable.

Is IMEXP for compound interest? It models a complex exponential, which can represent growth or rotation. Ordinary financial growth does not usually need a complex exponent; select the financial or real exponential function appropriate to the actual calculation.

Does IMEXP("5j") need a real coefficient? Zoho’s official example accepts this purely imaginary representation. The omitted real part is mathematically zero in that case; do not automatically generalize its text parser to every shortened syntax.

Can I use a cell rather than a literal? A single cell holding text such as 2+3j follows the documented argument meaning, and can be referenced directly. Use =IMEXP(A2), not =IMEXP("A2").

Will a spreadsheet calculate this identically on every platform? The mathematics is portable; app-specific parsing, argument separators, formatting and supported versions have not been established across all editions.

Quote complex literals, such as "2+3j", but do not quote a reference such as A2. Multi-argument formulas here use the semicolons printed by Zoho’s English help; follow your workbook’s formula editor if its syntax differs.

The official page lists broad error families: inspect spelling and quotation marks for #NAME!, input types for #VALUE!, and cell references for #REF!. Its generic list does not establish the exact error returned for each malformed complex input. Avoid applying unverified blank-to-zero or uppercase-suffix conversions.

• COMPLEX constructs a complex representation.
• IMABS extracts magnitude.
• IMREAL and IMAGINARY let you inspect the two coefficients.

Compare independently expected coefficients within a suitable numerical tolerance. Printed precision and simplified zero components can vary; the examples do not establish a universal string-serialization rule.