IMLN function

The IMLN function in Excel returns the natural logarithm of a given complex number. The natural logarithm of a complex number extends the logarithm function to the complex plane.

Syntax

IMLN(inumber)

Parameters

• inumber: The complex number for which you want to calculate the natural logarithm. The complex number can be entered as a string in the form a + bi or a + bj, where a is the real part and b is the imaginary part.

How It Works

For a complex number z = a + bi, the natural logarithm is calculated using the following formula:

$$\ln(z) = \ln|z| + i \cdot \theta$$

Where:

• $\ln|z|$ is the natural logarithm of the modulus (absolute value) of z, where $|z| = \sqrt{a^2 + b^2}$
• $\theta$ is the argument (phase angle) of z, which is $\tan^{-1}\left(\frac{b}{a}\right)$

Examples

1. Natural Logarithm of a Complex Number: To calculate the natural logarithm of the complex number 4 + 3i:

   =IMLN("4+3i")
   

   The result will be approximately 1.6094379124341 + 0.643501109i.

2. Natural Logarithm of a Real Number: If you use a real number such as 5 (with no imaginary part):

   =IMLN(5)
   

   The result will be approximately 1.6094379124341, which is the natural logarithm of 5.

3. Natural Logarithm of a Purely Imaginary Number: To calculate the natural logarithm of a purely imaginary number such as 0 + 2i:

   =IMLN("0+2i")
   

   The result will be 0.69314718055995 + 1.5707963267949i.

4. Using a Cell Reference: If cell A1 contains the complex number "3+4i", you can find its natural logarithm using:

   =IMLN(A1)
   

   The result will be approximately 1.6094379124341 + 0.92729521800161i.

Important Notes

• The IMLN function works only with valid complex numbers. If the input is not a valid complex number, Excel will return a #VALUE! error.
• For real numbers, the IMLN function behaves like the standard natural logarithm function.
• Complex numbers in Excel can use either i or j to represent the imaginary part.

Summary

The IMLN function in Excel calculates the natural logarithm of a complex number, extending the natural logarithm to the complex plane. This function is useful in advanced mathematical, engineering, and scientific applications where logarithms of complex numbers are needed.