 Login  |   Contact  |   About Us       Wednesday, October 4, 2023  ~ Trigonometric ~ Hyperbolic Functions ~ Inverse Functions ~ Complex Functions

TRIGONOMETRIC FUNCTIONS
Sine Function

We happen to know that a complex sine function can be calculated using complex exponential functions:

 sin z = 1/2i (eiz - e-iz) = 1/2i [ex(cos y + i sin y) - e-x(cos y - i sin y)] = sin x cosh y + i cos x sinh y

This being the case, we can implement a Sine method and add to the KMP library as follows:

public static Complex Sin(Complex c)
{
return new Complex(Math.Cosh(c.Imaginary) * Math.Sin(c.Real) +,
(Math.Sinh(c.Imaginary) * Math.Cos(c.Real));
}

Cosine Function

Similar to the case of the complex sine function, a complex cosine function can also be calculated using complex exponential functions:

 cos z = 1/2i (eiz + e-iz) = 1/2i [ex(cos y + i sin y) + e-x(cos y - i sin y)] = cos x cosh y - i sin x sinh y

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 Cos(Complex c)
{
return new Complex(Math.Cosh(c.Imaginary) * Math.Cos(c.Real) -,
(Math.Sinh(c.Imaginary) * Math.Sin(c.Real));
}

Tangent Function

A complex tangent function is computed using complex sine and cosine functions:

 tan z = sin z / cos z

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

public static Complex Tan(Complex c)
{
return Sin(c) / Cos(c);
}

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EIGENVALUE
SOLUTIONS...

> Rayleigh-Quotient Method

> Cubic Spline Method

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