This online Hyperbolic Functions Calculator computes hyperbolic functions of a complex number (variable). When typing the imaginary part of a complex number in the appropriate field of the calculator, make sure that the symbol ‘i‘, representing the imaginary unit, is adjacent to the numeric part without space.
Hyperbolic functions of one complex variable
This online calculator computes the following hyperbolic functions of a complex variable \(z=x+yi\), where \(x\) and \(y\) are real numbers. Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments.
• sinh() – hyperbolic sine function
The hyperbolic sine function of a real variable \(x\) is defined by:
$$ sinh (x) = \frac { { e }^{ x } – { e }^{ -x } } {2}.$$
Using Euler’s formula for complex numbers we can get the following expression for the hyperbolic sine function of a complex variable \(z\):
$$ sinh (z) = sinh(x) cos(y) + i cosh(x) sin(y).$$
• cosh() – hyperbolic cosine function
The hyperbolic cosine function of a real variable \(x\) is defined by:
$$ cosh (x) = \frac { { e }^{ x } + { e }^{ -x } } {2}.$$
We can get the following expression for the hyperbolic cosine function of a complex variable \(z\):
$$ cosh (z) = cosh(x) cos(y) + i sinh(x) sin(y).$$
• tanh() – hyperbolic tangent function
The hyperbolic tangent function is defined in the following way:
$$tanh(z)=\frac { sinh(z) }{ cosh(z) }.$$
• csch() – hyperbolic cosecant function
The hyperbolic cosecant function is defined in the following way:
$$ csch(z)=\frac { 1 }{ sinh(z) }.$$
• sech() – hyperbolic secant function
The hyperbolic secant function is defined in the following way:
$$ sech(z)=\frac { 1 }{ cosh(z) }.$$
• coth() – hyperbolic cotangent function
The hyperbolic cotangent function is defined in the following way:
$$ coth(z)=\frac { 1 }{ tanh(z) }.$$
The inverse hyperbolic functions are the inverse functions of the hyperbolic functions. Specifically, they are the inverses of the hyperbolic sine, cosine, tangent, cotangent, cosecant and secant functions. As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic, except at a finite number of points. For such a function, it is common to define a principal value, which is a single valued analytic function which coincides with one specific branch of the multivalued function. For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function.
In what follows, for the square root, the principal value is defined as the square root that has a positive real part. Similarly, the principal value of the logarithm is defined as the value for which the imaginary part has the smallest absolute value. Considering the square root and the logarithm to be the principal values, we have the following expressions for the inverse hyperbolic functions.
$$arsinh(z) = \ln \left( \sqrt{1+z^2} + z \right),$$
$$arcosh(z) = \ln \left( \sqrt{1+z} \sqrt{1-z} + z \right),$$
$$artanh(z) = \frac{1}{2}\ln \left(\frac{1 + z}{1 – z}\right),$$
$$arcoth(z) = \frac{1}{2}\ln \left(\frac{z + 1}{z – 1}\right),$$
$$arsech(z) = \ln \left(\sqrt{\frac{1}{z} + 1} \sqrt{\frac{1}{z} – 1} + \frac{1}{z} \right),$$
$$arcsch(z) = \ln \left(\sqrt{\frac{1}{z^2} + 1} + \frac{1}{z} \right).$$
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