Complex trigonometric function . In Mathematics and, more specifically, in Mathematical Analysis are the trigometric functions whose arguments and results are complex numbers . It is based on the relationship between the trigonometric and exponential representations of complex numbers. It is of vital importance in the formulation of other complex functions such as: the logarithm , complex exponential functions, and others; facilitating the extension of the trigonometry concept to complex angles.
Summary
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- 1 Definitions
- 1 Circular trigonometric functions
- 2 Hyperbolic trigonometric functions
- 2 Importance
- 3 See also
- 4 Sources
Definitions
Let the complex numbers x = a + bi and y = c + di define the following functions F (x) = y with their restrictions as complex trigonometric functions because their arguments and / or results are complex:
Circular trigonometric functions
The well-known circular trigonometric functions with real argument x :
- sin (x)
- cos (x)
- so (x)
- cot (x)
They can be evaluated on complex numbers using the complex exponential function with the following expressions:
- File: Complex exponential cotangent definition.gif
Hyperbolic trigonometric functions
The well-known circular trigonometric functions with real argument x :
- sh (x)
- ch (x)
- th (x)
- coth (x)
They can be evaluated on complex numbers using the complex exponential function with the following expressions:
- File: Complex exponential hyperbolic cotangent definition.gif
Importance
The law of Euler establishes an important link between trigonometry and calculus, to the associate exponentiation of the imaginary part complex with the representation of a complex whose magnitude is 1. Trigonometry complex not only extends the reach of traditional trigonometric relationships but which gives it greater significance in the calculation.
