Complex numbers complete the description of numbers as we know, by without losing or spoiling any of the known data in the physical world. It is just a developed notation to handle the extra numbers came along with the solutions of equations, and named to be the "so called" physically meaningless solutions.
The notation used to represent the most general form of complex number is, $$ z = a + ib \,(or)\, x+iy $$ where "i" is the imaginary root i.e. the root of "-1". With this new notation, any known number in our Nature, can be written in this form.
Once we expand the number system with this new notation, it eventually expands all the fundamental definitions used by those numbers in any field. It leads to subsequent changes in all of the functions that is defined over the real numbers. For example, we can analyze the effect in our usual well behaved functions such as trigonometric function, exponential functions, etc.
We need to remember one thing that, the new real functions we are going to define in a complex domain is simply just the extension of the foreknown concepts and they are all just axiomatic definitions. So, it is not possible to ask for the proof of these definitions!
For example, in the polar form, a complex number is denoted by $$ z = r\,e^{i\theta} $$ where $\theta = \theta_p \,+\, 2\pi\,k$ k= 1,2,3,..
$\theta_p$ is called the principle angle measured from the positive x-axis.
Now, we extend this concept of trigonometric functions into complex functions by replacing the real x-values with new complex numbers. It is achieved with the help of the handful tool i.e. series expansions of all powers of x.
Since complex part is in the form of addition, all these powers just adds extra terms into our real expansion series. So that, the essence of old functional forms are not affected in anyway due to this new definition transformation, except that it was just incorporated in a larger domain.
Using Euler's formula, the general form of a complex number can be denoted as, $$ z\, = \, re^{i\theta} = r (cos{\theta} + i sin{\theta}) $$
From Euler's formula, we tend to write sin($\theta$) and cos($\theta$) in terms of exponential functions, where each exponential form is used to represent a complex number.
Euler's formula also gives, $$ e^{-i\theta}\, = \,cos(\theta)\, -\, i\,sin(\theta)\,$$ Thus we get,
$$ cos(\theta) \,=\, \frac{e^{i\theta} + e^{-i\theta}}{2} $$ and $$ sin(\theta) \, =\, \frac{e^{i\theta} - e^{-i\theta}}{2i} \,$$
As the right hand side of this equation only deals with exponentials, if we replace the $\theta$ with z - complex number, the definition of sine function expands to wider regions which includes all the complex numbers.
And so, a new name and definition is given, where the independent variable $\theta$ is replace by the complex number "z". And they are hyperbolic trigonometric functions.
The complex number "iy" is used in the euler equation to give,
$$ e^{\{i(iy)\}}\, = \,e^{-y} \,= \,cos(iy)\, + \,i \,sin(iy)\, $$ Similarly, $$ e^{\{-i(iy)\}}\, = \,e^y \,= \, cos(iy) \, - \, i\, sin(iy) \,$$
We get, $$ cos(iy) \,=\, \frac{e^y\, + \,e^{-y}\,}{2} = cosh(y) $$
We choose this cos(iy) as cosh(y) since it has the similar form of cosine in Euler formula.
But sine is defined from, $$ sin(iy) = \,\frac{e^{-y}\,-\,e^{y}\,}{2i} = - (\frac{e^{y} - e^{-y}}{2i} = -i sinh(y)$$ so that the both structures of the equation will look similar.
As we ourselves changed the basic definition, we cannot expect the same results of usual trigonometry in here. For example, the maximum value of sine and cos is equal to one when dealing real numbers, but with complex number it can have any value
That is all we need to know about definitions.. now we can proceed further to define all other identities from this basic concept and all other things can be sought out from those definitions.
The notation used to represent the most general form of complex number is, $$ z = a + ib \,(or)\, x+iy $$ where "i" is the imaginary root i.e. the root of "-1". With this new notation, any known number in our Nature, can be written in this form.
Once we expand the number system with this new notation, it eventually expands all the fundamental definitions used by those numbers in any field. It leads to subsequent changes in all of the functions that is defined over the real numbers. For example, we can analyze the effect in our usual well behaved functions such as trigonometric function, exponential functions, etc.
We need to remember one thing that, the new real functions we are going to define in a complex domain is simply just the extension of the foreknown concepts and they are all just axiomatic definitions. So, it is not possible to ask for the proof of these definitions!
For example, in the polar form, a complex number is denoted by $$ z = r\,e^{i\theta} $$ where $\theta = \theta_p \,+\, 2\pi\,k$ k= 1,2,3,..
$\theta_p$ is called the principle angle measured from the positive x-axis.
Now, we extend this concept of trigonometric functions into complex functions by replacing the real x-values with new complex numbers. It is achieved with the help of the handful tool i.e. series expansions of all powers of x.
Since complex part is in the form of addition, all these powers just adds extra terms into our real expansion series. So that, the essence of old functional forms are not affected in anyway due to this new definition transformation, except that it was just incorporated in a larger domain.
Using Euler's formula, the general form of a complex number can be denoted as, $$ z\, = \, re^{i\theta} = r (cos{\theta} + i sin{\theta}) $$
From Euler's formula, we tend to write sin($\theta$) and cos($\theta$) in terms of exponential functions, where each exponential form is used to represent a complex number.
Euler's formula also gives, $$ e^{-i\theta}\, = \,cos(\theta)\, -\, i\,sin(\theta)\,$$ Thus we get,
$$ cos(\theta) \,=\, \frac{e^{i\theta} + e^{-i\theta}}{2} $$ and $$ sin(\theta) \, =\, \frac{e^{i\theta} - e^{-i\theta}}{2i} \,$$
As the right hand side of this equation only deals with exponentials, if we replace the $\theta$ with z - complex number, the definition of sine function expands to wider regions which includes all the complex numbers.
And so, a new name and definition is given, where the independent variable $\theta$ is replace by the complex number "z". And they are hyperbolic trigonometric functions.
The complex number "iy" is used in the euler equation to give,
$$ e^{\{i(iy)\}}\, = \,e^{-y} \,= \,cos(iy)\, + \,i \,sin(iy)\, $$ Similarly, $$ e^{\{-i(iy)\}}\, = \,e^y \,= \, cos(iy) \, - \, i\, sin(iy) \,$$
We get, $$ cos(iy) \,=\, \frac{e^y\, + \,e^{-y}\,}{2} = cosh(y) $$
We choose this cos(iy) as cosh(y) since it has the similar form of cosine in Euler formula.
But sine is defined from, $$ sin(iy) = \,\frac{e^{-y}\,-\,e^{y}\,}{2i} = - (\frac{e^{y} - e^{-y}}{2i} = -i sinh(y)$$ so that the both structures of the equation will look similar.
As we ourselves changed the basic definition, we cannot expect the same results of usual trigonometry in here. For example, the maximum value of sine and cos is equal to one when dealing real numbers, but with complex number it can have any value
That is all we need to know about definitions.. now we can proceed further to define all other identities from this basic concept and all other things can be sought out from those definitions.
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