In the following chapters, we will see that, in order to define complex functions and their features, we can extend some properties of real functions; but, in some cases the mere extension will not be enough.

We start with the following:

Consider ${\mathcal {D}}\subset \mathbb {C}$. A complex-valued function is a function:

$f:{\mathcal {D}}\rightarrow \mathbb {C} .$

We see that $\forall \,z=x+iy\in {\mathcal {D}}$ the function $f$ can be decomposed as:

$f(z)=u(z)+iv(z)\ ,$

where the functions

$u,v$ are real-valued functions. In particular, we define:

$u=Re(f)\;\&\;v=Im(f)\ .$

Alternatively, we can use the isomorphism between

$\mathbb {C}$ and

$\mathbb {R} ^{2}$ and look at

$u$ and

$v$ as two variables real functions:

$f(z)=f(x+iy)=u(x,y)+iv(x,y)\ .$

In fact, by calling ${\mathcal {\tilde {D}}}\subset \mathbb {R} ^{2}$, we have that: $u,v:{\mathcal {{\tilde {D}}\rightarrow \mathbb {R} }}\ .$

Now we define some complex functions as extensions of known real functions:

**Trigonometric Functions**: We recall that, thanks to the Euler's formula, we can write:

$\cos x={\frac {e^{ix}+e^{-ix}}{2}}\ ,$

So, the natural extension to the complex field is:

$\cos z={\frac {e^{iz}+e^{-iz}}{2}}\ ,$

Analogously, we have that:

$\sin z={\frac {e^{iz}-e^{-iz}}{2i}}\ ,$

by setting

$z=x+iy$ we see that:

$\sin(z)=sin(x+iy)=$

$={\frac {e^{i(x+iy)}+e^{-i(x+iy)}}{2i}}=$

$={\frac {e^{-y}e^{ix}-e^{y}e^{-ix}}{2i}}\ .$

We underline that $\mid e^{\pm ix}\mid =e^{\pm ix}(e^{\pm ix})^{*}=e^{\pm ix}e^{\mp ix}=1.$ All the complex exponentials of the form $e^{if(x)}{\mbox{, where f(x) can be also a constant function,}}$ are called *phases* and have unitary modulus.

The following properties (simple to prove) are valid:

- $\cos ^{2}z+\sin ^{2}z=1\ ,$
- $\cos(iy)=\cosh y\ ,$
- $\sin(iy)=i\sinh y\ ,$
- $\cos(z)=\cos(x+iy)=\cos x\cosh y-i\sin x\sinh y\ ,$
- $\sin(z)=\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y\ .$

**Logarithm function** in $\mathbb {C}$: In order to define the complex logarithm, we use the polar form of complex numbers. So, we have:

$\log(z)=\log(\mid z\mid e^{i\theta })=\log(\mid z\mid e^{iArg(z)})=$

$=\log(\mid z\mid )+iArg(z)\ .$

Actually, we have to keep in mind that complex numbers are defined with a certain ambiguity in their polar form, so we have that:

$\log(z)=\log(\mid z\mid )+i(Arg(z)+2k\pi ),\;k\in \mathbb {Z} \ .$

The following properties are true:

- $\log(-1)=i(\pi +2k\pi )\ ,$
- $\log(i)=i({\frac {\pi }{2}}+2k\pi )\ ,$
- $\forall \,z_{1},z_{2}\in \mathbb {C} {\mbox{ ,we have that }}\log(z_{1}z_{2})=\log(z_{1})+\log(z_{2})+2k\pi i\ ,$
- $\forall \,z_{1},z_{2}\in \mathbb {C} {\mbox{ ,we have that }}\log({\frac {z_{1}}{z_{2}}})=\log(z_{1})-\log(z_{2})+2k\pi i\ .$