IMSINH function

The IMSINH function in Excel returns the hyperbolic sine of a complex number.

Syntax

IMSINH(inumber)

Parameters

• inumber: The complex number for which you want to calculate the hyperbolic sine. You can input the complex number in the form of a text string such as a + bi or a + bj, where a is the real part and b is the imaginary part. Alternatively, you can use a cell reference that contains a complex number.

How It Works

The IMSINH function calculates the hyperbolic sine of a complex number. The formula for the hyperbolic sine of a complex number $z = a + bi$ is:

$$\text{sinh}(z) = \frac{e^{z} – e^{-z}}{2}$$

Where:

• $a$ is the real part of the complex number,
• $b$ is the imaginary part,
• $e^z$ and $e^{-z}$ represent the exponential functions of the complex number.

Examples

1. Hyperbolic Sine of a Complex Number: To calculate the hyperbolic sine of the complex number 2 + 3i:

   =IMSINH("2+3i")
   

   The result will be a complex number representing the hyperbolic sine of 2 + 3i.

2. Using Cell References: If cell A1 contains the complex number "1+2i", you can calculate the hyperbolic sine using:

   =IMSINH(A1)
   

   The result will be the hyperbolic sine of the complex number in A1.

3. Purely Imaginary Number: For a purely imaginary number such as "0 + 4i", you can calculate the hyperbolic sine as:

   =IMSINH("0+4i")
   

   The result will be the hyperbolic sine of 4i, which is a complex number.

Important Notes

• The IMSINH function only works with valid complex numbers. If the input is not a valid complex number, Excel will return a #VALUE! error.
• Complex numbers can be written using either i or j for the imaginary part.
• The result of the IMSINH function will always be a complex number.

Summary

The IMSINH function in Excel is used to calculate the hyperbolic sine of a complex number. This function is valuable for advanced mathematical, engineering, and scientific applications that involve hyperbolic functions for complex numbers.