# Documentation/Calc Functions/IMLN

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IMLN

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## Summary:

Calculates the logarithm to the base e or natural logarithm of a complex number. e is a mathematical constant known as Euler's number, which has a value of approximately 2.71828.

## Syntax:

IMLN(ComplexNumber)

## Returns:

Returns a text string giving the complex number that is the principal natural logarithm of the complex number passed as argument. The imaginary part of the principal natural logarithm lies in the range -π < x ≤ π.

## Arguments:

ComplexNumber is a string (in quotation marks) representing a complex number in the form a+bi or a+bj, or a reference to a cell containing that string, which is the complex number for which the natural logarithm is to be calculated. a and b are real numbers.

• If ComplexNumber does not have an imaginary part (b equal to 0), the real part can be passed to IMLN as a number rather than a string.
• If both real and imaginary parts of ComplexNumber are 0, IMLN reports an invalid argument error (Err:502).
• If ComplexNumber is not a valid complex number, IMLN reports an invalid argument error (Err:502).

## Additional details:

### Details specific to IMLN function

For a complex number of the form a+bi, the natural logarithm is given by the following equation:

IMLN uses an alternative formulation for the imaginary part based on an equivalent inverse cosine function call instead of the inverse tangent function call, along with a necessary check to ensure that the correct sign is applied to the imaginary part of the result.

For more information, visit Wikipedia’s Complex logarithm page.

### General information about Calc's handling of complex numbers For convenience, the information in this subsection is repeated on all pages describing functions that manipulate complex numbers.

• A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is a symbol for the imaginary unit, such that i2=−1. a is known as the real part of the complex number and b is known as its imaginary part.
• In some disciplines using i to indicate the imaginary unit may be undesirable and, in such cases, the symbol j may be used instead. The functions provided by Calc to process complex numbers can use either i or j.
• For additional general background information, visit Wikipedia’s Complex number page.
• Within Calc, a complex number is represented as a string expression of the form "a+bi" or "a+bj", where a and b are real numbers. If the complex number happens to be a real number (b=0), then it can be represented as either a string expression or a numeric value. Complex number string expressions should not contain any space characters. Examples of valid Calc complex numbers include "2", -6, "2+3i", "3j", "12.34+56.78i", and "1e+23-1e-22j". Calc would not recognize the strings "1+j2", "i+1", and "i+j" as valid complex numbers.
• The representation of the complex numbers in strings can have up to 15 significant digits and scientific style is automatically used if needed (visit Wikipedia’s Scientific notation page for more information). The decimal delimiter in complex number string expressions is always a dot, irrespective of locale setting.

## Examples:

Formula Description Returns
=IMLN("1.2-3.4j") Here the function calculates the natural logarithm of a complex number that has non-zero real and imaginary parts. 1.28247467873077-1.23150371234085j
=IMLN("-i") Here the function calculates the natural logarithm of a complex number that has no real part. The imaginary part of the result is equal to -π/2 -1.5707963267949i
=IMLN(-1) Here the function calculates the natural logarithm of a complex number that has no imaginary part. The imaginary part of the result is equal to π. 3.14159265358979i
=IMLN("-1e20-i") Here the function calculates the natural logarithm of a complex number that has non-zero real and imaginary parts. The imaginary part of the result is close (but not equal) to -π. 46.0517018598809-3.14159265358979i
=IMLN(F1) where cell F1 contains the complex number 2-3j. Here the function calculates the natural logarithm of a complex number that is entered in a cell. 1.28247467873077-0.982793723247329j

IMLN