IMCOSH function

The IMCOSH function in Excel returns the hyperbolic cosine of a given complex number. The hyperbolic cosine is similar to the regular cosine function but is based on hyperbolic functions, which are used in various mathematical fields, including engineering and physics.

Syntax

IMCOSH(inumber)

Parameters

• inumber: The complex number for which you want to calculate the hyperbolic cosine. The complex number can be provided 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 hyperbolic cosine is calculated using the formula:

$$\cosh(z) = \frac{e^z + e^{-z}}{2}$$

Where:

• a is the real part of the complex number.
• b is the imaginary part of the complex number.
• e^z is the exponential function.

Examples

1. Basic Example: To calculate the hyperbolic cosine of the complex number 1 + 2i:

   =IMCOSH("1+2i")
   

   The result will be approximately -0.64214812471552 + 1.0686074213828i.

2. Real Number: If you want to calculate the hyperbolic cosine of a real number like 3 (no imaginary part):

   =IMCOSH(3)
   

   The result will be 10.06766199578, as this is the hyperbolic cosine of the real number 3.

3. Purely Imaginary Number: For a purely imaginary number such as 0 + 2i:

   =IMCOSH("0+2i")
   

   The result will be -1.56562583532.

4. Using a Cell Reference: If cell A1 contains the complex number "2+3i", you can find its hyperbolic cosine using:

   =IMCOSH(A1)
   

   The result will be approximately -3.72454550492 + 0.511822569987i.

Important Notes

• The IMCOSH function works only with valid complex numbers. If the input is not a valid complex number, Excel will return a #VALUE! error.
• The result of the IMCOSH function is a complex number that may have both real and imaginary parts.
• For real numbers, the IMCOSH function behaves like the standard hyperbolic cosine (COSH) function, returning the hyperbolic cosine of the real number.
• Both i and j can be used to denote the imaginary unit in Excel’s complex number functions.

Summary

The IMCOSH function in Excel calculates the hyperbolic cosine of a complex number. This is especially useful in advanced mathematics, engineering, and physics, where hyperbolic functions and complex numbers are commonly used.