Hyperbolic Tangent of Complex Number - ProofWiki
2018年9月15日 · Let $i$ be the imaginary unit. Then: Formulation 1 $\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$ Formulation 2 $\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$ Formulation 3
Hyperbolic functions - Wikipedia
In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
Inverse hyperbolic functions - Wikipedia
Graphs of the inverse hyperbolic functions The hyperbolic functions sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
We also carefully define the corresponding single-valued principal values of the inverse trigonometric and hyperbolic functions following the conventions of Abramowitz and Stegun (see ref. 1). 1. The inverse trigonometric functions: arctan and arccot. We begin by examining the solution to the equation. sin w 1.
Tangent of Complex Number - ProofWiki
2018年9月15日 · Theorem. Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: Formulation 1. $\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$ Formulation 2. $\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$ Formulation 3.
Hyperbolic functions of complex numbers - simply.science
The hyperbolic functions are periodic w.r.t. the imaginary component with period 2πi for sinh, cosh, sech & cosech; and with period πi for tanh and coth. Hyperbolic functions in the complex plane z representing a complex number, the functions sinh z and cosh z are said to be holomorphic or analytic (see below figures).
complex analysis - Real and imaginary part of $\tan (a+bi ...
When dealing with complex trigonometric functions, it is quite natural to ask how the real/imaginary part of $\tan (a+bi)$ can be expressed using $a$ and $b$. Of course, since $\tan z$ and $\tanh z$ are tightly linked for complex variables, we could derive the real/imaginary part for hyperbolic tangent from the corresponding results for $\tanh ...
Introduction to the Hyperbolic Tangent Function - Wolfram
Definition of the hyperbolic tangent function for a complex argument. In the points , where has zeros, the denominator of the last formula equals zero and has singularities (poles of the first order). Here are two graphics showing the real and imaginary parts of the hyperbolic tangent function over the complex plane.
Tanh: Get the hyperbolic tangent—Wolfram Documentation
Tanh is the hyperbolic tangent function, which is the hyperbolic analogue of the Tan circular function used throughout trigonometry. Tanh [ α ] is defined as the ratio of the corresponding hyperbolic sine and hyperbolic cosine functions via .
Tanh Calculator
This tanh calculator is the ultimate tool you need to deal with the hyperbolic tangent function! In what follows, we will explain what tanh is in math and how to calculate both the inverse and the derivative of tanh. We won't forget to give you extra tips on how to use this hyperbolic tangent calculator most efficiently. Let's go!
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