How to use the IMCSCH function
What is the IMCSCH function?
The IMCSCH function calculates the hyperbolic cosecant 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 a complex number?
A complex number contains a real and imaginary value, they let you for example solve equations with no real solutions like x2 + 1 = 0Â The chart above shows x2 + 1 = 0 and it never touches the x-axis.
However, mathematicians discovered the imaginary number i that solves equations into the complex plane.
What is the hyperbolic cosecant?
Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them.
Trigonometric functions use a circle, while hyperbolic functions use a hyperbola. The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola. The chart below shows a circle containing the trigonometric functions.
Table of Contents
1. IMCSCH Function Syntax
IMCSCH(inumber)
2. IMCSCH Function Arguments
| inumber | Required. A complex number in x+yi or x+yj text format. |
3. IMCSCH function example
The image above demonstrates a formula in cell B28 that calculates the hyperbolic cosecant of a complex number specified in cell B25.
Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.
The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.
Formula in cell D3:
The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.
Complex number 1+2i is the light blue line in the first quadrant. The hyperbolic cosecant of 1+2i is the green line displayed in the third quadrant.
3.1 Explaining formula
Step 1 - Populate arguments
IMCSCH(inumber)
becomes
IMCSCH(B25)
Step 2 - Evaluate the IMCSCH function
IMCSCH(B25)
becomes
IMCSCH("1+2i")
and returns
-0.221500930850509-0.6354937992539i
4. How is the IMCSCH function calculated in detail?
The hyperbolic cosecant of a complex number is calculated like this:
csch(x + yi) = sinh(x)*cos(y) - icosh(x)*sin(y) / (sinh2(x)*cos2(y)+cosh2(x)*sin2(y))
For example, C=1+2i
csch(1 + 2i) = sinh(1)*cos(2) - icosh(1)*sin(2) / (sinh2(1)*cos2(2)+cosh2(1)*sin2(2))
becomes
csch(1 + 2i) = ( 1.1752011936438*-0.416146836547142 - i1.54308063481524*0.909297426825682 ) / (1.38109784554182*0.173178189568194+2.38109784554182*0.826821810431806)
becomes
csch(1 + 2i) = ( -0.489056259041293 - 1.40311925062204i ) / 2.20791965597362
equals
csch(1 + 2i) = -0.221500930850509-0.6354937992539i
5. IMCSCH function not working
The IMCSCH function returns a #NUM! error if the provided argument is not a valid complex number.
The IMCSCH function returns a #VALUE! error if the provided argument is a boolean value.
Useful links
IMCSCH function - Microsoft
Hyperbolic Cosecant of Complex Number
Hyperbolic functions
Functions in 'Engineering' category
The IMCSCH function function is one of 42 functions in the 'Engineering' category.
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