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"Complex Inverse Trigonometric Functions"
Complex Inverse Hyperbolic Methods

Specify New Function Values = { , }

[ Initial Inverse Hyperbolic Values specified: {2,3} ]

Inverse Hyperbolic Sine Function

In general, the complex inverse hyperbolic functions can be expressed in terms of the complex square root and logarithmic function. The complex inverse Sinh function can be expressed in the following equation:

arcsin h(z) = log( z + √(1 + z)2)

This being the case, an Asinh method can be implemented and added to the KMP library as follows:

           public static Complex Asinh(Complex c)
           return log(c + Sqrt(1.0 * c));


Inverse Hyperbolic Cosine Function

Similar to the case of the complex Asinh function, a complex Acosh function can also be calculated using complex exponential functions. The complex inverse Cosh function can be represented by:

arcos h(z) = log( z + √(z)2 - 1)

Using the above equation we can add a new method to the "Complex class". (Note - Complex class is an integral part of our software library and is not shown in these examples):

           public static Complex Acosh(Complex c)
           return log(c + Sqrt(c * c - 1.0));


Inverse Hyperbolic Tangent Function

The complex inverse Tanh function has the following relation:

arctan h(z) = 1/2 [log(1 + z) - log(1 - z)]

The following complex Tangent method is added to the Complex class:

           public static Complex Atan(Complex c)
             return 0.5 * [log(1 +c) - log(1 - c)];

Testing the Inverse Hyperbolic Methods

In order to test the Inverse Hyperbolic methods as defined above, a new TestInverse Hyperbolic() static method has been added and executed. Supporting code and methods are not shown.

           static void TestInverseHyperbolic();
                 ListBox1.Items.Add(" c = " + c);
                 ListBox2.Items.Add(" " + Complex.Asinh(c));
                 ListBox3.Items.Add(" " + Complex.Acosh(c));
                 ListBox4.Items.Add(" " + Complex.Atanh(c));

We first defined a set of data points (2,3). The user can manipulate values and try variations on new points themselves.

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