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Inverse Sine Function

Although we did not include a practical application using complex inverse trigonometric functions, we feel it follows logic presented before and may be useful to our readers. Inverse trigonometric functions can be expressed in terms of the complex square root and logarithmic functions.

Inverse Sine Function

A complex inverse sine function can be expressed as follows::

arcsin z = - i log(iz + √1 - z2)

Using this relationship, we created the following method:

           public static Complex Asin(Complex c)
              return - I * Log(Sqrt(1 - c * c) + I * c);

Inverse Cosine Function

Similar to the case of the complex inverse sine function, a complex inverse cosine function can also be represented by:

arccos z = - i log(z + i √1 - z2)

Using the above equation we added a new method to the "Complex class". (Note - Complex class is not shown in these examples):

           public static Complex Acos(Complex c)
              return - I * Log( I * Sqrt(1 - c * c) + c);

Inverse Tangent Function

The complex inverse tangent function has the following relation:

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

Adding a complex inverse Tangent method to the Complex class:

           public static Complex Atan(Complex c)
             return 0.5 * I * (Log(1 - I * c) - Log(1 + I * c));

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