Documentation/Calc Functions/IMCSCH

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IMCSCH

Summary:

Calculates the hyperbolic cosecant of a complex number. The hyperbolic cosecant is the reciprocal of the hyperbolic sine.

Syntax:

IMCSCH(ComplexNumber)

Returns:

Returns a text string representing the complex number that is the hyperbolic cosecant of the complex number passed as argument.

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, whose hyperbolic cosecant is to be determined. 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 IMCSCH as a number rather than a string.
• If ComplexNumber is not a valid complex number, IMCSCH reports an invalid argument error (Err:502).

Details specific to IMCSCH function

For a complex number of the form a+bi, IMCSCH implements the following equation:

However, IMCSCH determines whether b=0 and in this case simply returns 1/SINH(a).

General information about Calc's handling of complex numbers

 Note: 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:

 Note: For complex numbers in Calc, the decimal point is used as decimal separator.
Formula Description Returns
=IMCSCH("1.2-3.4j")

Calculate the hyperbolic cosecant of a complex number that has non-zero real and imaginary parts.

-0.622647059064193​-​0.197415259991078j
=IMCSCH("-i")

Calculate the hyperbolic cosecant of a complex number that has no real part. The result has no real part either.

1.18839510577812i
=IMCSCH(-1)

Calculate the hyperbolic cosecant of a complex number that has no imaginary part. The result has no imaginary part either.

-0.850918128239322
=IMCSCH(F1)

where cell F1 contains the complex number 2-3j.

Calculate the hyperbolic cosecant of a complex number that is entered in a cell.

-0.27254866146294​+​0.0403005788568915j

IMCSCH