What is "the scalar-valued functions from $\mathbb R_+$ to $\mathbb C$"? Example from book Also see the update below for clarification.
I have the following:

Let $a$, $b \in \mathbb R$ and write $c=a+ib\in\mathbb C$. Consider the scalar-valued functions $\exp_c$, $\sin_c$ and $\cos_c$ from $\mathbb R_+$ to $\mathbb C$ defined by
\begin{align}
\exp_c(t)&=e^{ct} \tag 1\\
\sin_c(t)&=\sin(ct) \tag 2\\
\cos_c(t)&=\cos(ct)\tag 3\\
\end{align}
$\forall t\in \mathbb R_+$.
Then the Laplace transform of $\exp_c$ is given by
$$
\mathcal L\{\exp_c\}(s)=\int_0^\infty e^{-(s-c)t} \, dt = \frac{1}{s-c}
$$
for all $s\in \mathbb C$ with $\Re(s)> \Re (c) =a$.

My questions are regarding $(1)-(3)$.
I don't follow the equality. It says "the scalar-valued functions $\exp_c$, $\sin_c$ and $\cos_c$ from $\mathbb R_+$ to $\mathbb C$"
But in the LHS in $(1)-(3)$ we have for all $t\in \mathbb R_+$
\begin{align}
\exp_c:\mathbb R \to \mathbb C \tag 4\\
\sin_c:\mathbb R \to \mathbb C \tag 5\\
\cos_c:\mathbb R \to \mathbb C \tag 6\\
\end{align}
But in the RHS in $(1)-(3)$ we have for all $c\in\mathbb C$
\begin{align}
\exp:\mathbb C \to \mathbb C \tag 7\\
\sin:\mathbb C \to \mathbb C \tag 8\\
\cos:\mathbb C \to \mathbb C \tag 9\\
\end{align}
What have I misapprehend  here?
For a "scalar-valued function" I think of a function $\mathbb R\to \mathbb R$ or $\mathbb C \to \mathbb R$. So what is the meaning of "scalar-valued functions" here?
Update:
Regarding the comments. I don't follow the meaning of "for a fixed $c\in \mathbb C$".
If we have "the scalar-valued functions $\exp_c$, $\sin_c$ and $\cos_c$" I'm thinking of functions of the form $f:\mathbb R \to \mathbb R$, i.e. $f(t)$ or $g:\mathbb C \to \mathbb R$, i.e. $g(z)$. And for a complex-valued function (of a real variable) $h: \mathbb R \to \mathbb C$, i.e. $h(t) = u(t)+iv(t)$.
Can someone shed some light here regarding this?
 A: It is allowed to consider complex-valued functions
$$h:\quad{\mathbb R}_{\geq0}\to{\mathbb C},\qquad t\mapsto h(t)\ .$$
For each $t\in{\mathbb R}_{\geq0}$ the value $h(t)$ is an element of ${\mathbb C}$, i.e., a bona fide complex number. This number is a single mathematical object. E.g.,  if $h(\pi)$ is important in the sequel  we could write $h(\pi)=: p$ without calculating anything numerically. Of course each value $h(t)$ has a real and an imaginary part; but it is not necessary to talk about these, or to invent special names  $u$, $v$ for them.
Now, in the context of Laplace transform, your author has the feeling that certain special functions ${\mathbb R}_{\geq0}\to{\mathbb C}$ will be important, and will recur all the time. Therefore he invents particular names for them, only valid in his book. Since these functions are related to well known functions from Calculus 101 the author forms their names along the  known names.
These are not only two or three special functions, but an infinity of them, parametrized by an auxiliary complex variable $c$. In this way we arrive at the family $\bigl(\exp_c\bigr)_{c\in{\mathbb C}}$ of functions
$$\exp_c:\quad{\mathbb R}_{\geq0}\to{\mathbb C},\qquad t\mapsto e^{c\,t}\ .$$
Here it is assumed that the reader is familiar with the general definition
$$e^z:=\sum_{k=0}^\infty{z^k\over k!}\qquad(z\in{\mathbb C})\ .$$
