The IMEXP function calculates the exponential of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

What is the exponential e?
Exponential e is an irrational number also called Euler's number. It is approximately 2.718

What is an irrational number?
It is a number that can't be expressed as a simple fraction, in other words, the number of decimals are infinite. Other examples are √2 and π.

Why is e called Euler's number?
Leonard Euler is the first one to use the exponential e in the 18-th century. e is also known as the base of natural logarithms which are logarithms to the base of e.

Why is e known as the base of natural logarithms?
The number e is known as the base of natural logarithms because the natural logarithm function is the inverse of the natural exponential function.
x = e^{ln x} or x = ln e^x

In what applications are complex logarithms useful?
Many fields of mathematics and scientific disciplines use logarithms extensively.

• compound interest formulas
• exponential decay formulas

What is the exponential form?
Calculations with trigonometric functions and exponential functions of complex numbers become simpler with this form. It also demonstrates the connection between complex numbers and cyclical phenomena, such as waves and oscillations.

Z = re^(iθ)

1. IMEXP Function Syntax

IMEXP(inumber)

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2. IMEXP Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMEXP function example

The image above demonstrates a formula in cell B28 that calculates the exponential of a complex number specified in cell B25.

Formula in cell B28:

The chart above demonstrates the complex plane, the y-axis the the imaginary axis and the x-axis is the real axis.

Complex number 2+i is the light blue line in the first quadrant. The exponential of 2+i is the green line also in the first quadrant.

3.1 Explaining formula

Step 1 - Populate arguments

IMEXP(inumber)

becomes

IMEXP(B25)

Step 2 - Evaluate the IMEXP function

IMEXP(B25)

becomes

IMEXP("2+i")

and returns

-3.99232404844127+6.21767631236797i

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4. How is the exponential of a complex number calculated in detail?

The exponential of a complex number is calculated like this:

C = x + yi

IMEXP(C) = e^(x+yi) = e^xe^yi = e^x(cos y + isin y)

For example, if C = 2+i then

IMEXP(C) = e⁽²⁺ⁱ⁾ = e²eⁱ = e²(cos 1 + isin 1)

e²(cos 1 + isin 1) = e² * cos 1 + ie² *sin 1

becomes

7.38905609893065*cos 1 + i7.38905609893065*sin 1

becomes

7.38905609893065*0.54030230586814 + i7.38905609893065*0.841470984807897

equals

3.99232404844127 + 6.21767631236797i

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5. The exponential function of a complex number produces a wave

The image above demonstrates how the exponential function of a complex number results in a wave shown in the chart.

C=x+yi

The imaginary part starts with 0 (zero) in cell C25 and is increased by (1/2)π  or 45 degrees for each cell below. The real number is always 0 (zero) in this example.

The IMEXP function calculates the exponential in cells D25 and below for each complex number and the result shows the real number and the imaginary number oscillating back and forth.

The chart shows the real numbers in orange and the imaginary numbers in light blue, the real axis displays radians in fractions of pi in steps of (1/2)π.

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Useful links

IMEXP function - Microsoft
Exponential Form of a Complex Number
Euler's formula - Wikipedia

Functions in 'Engineering' category

The IMEXP function function is one of many functions in the 'Engineering' category.

Excel function categories

Excel categories

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