How to find the roots of a linear combination of transcendental functions. I have run across a situation where I need the roots of a function of the following form:
$$f(t) = A sin(wt) - Be^{-Ct}$$
I started to try a series approach, but that quickly got into unfamiliar territory. Is the series route the way to go? If so, could anyone direct me to some material on the subject? I'm not really sure what I'm looking for.
 A: You want to find the zeros of function $$f(t) = A \sin(wt) - Be^{-Ct}$$ To simplify, let
$$x=wt\qquad \qquad a=\frac B A\qquad \qquad b=\frac C w$$ and consider
$$g(x)=\sin (x)-a\, e^{-b x}$$ where all coefficients are positive.
For the first positive root,we have
$$g(0)=-a  <0 \qquad\qquad  g'(0)=1+a b >0\qquad \qquad g''(0)=-a b^2 <0$$ So, by Darboux theorem, using Newton method, we shall reach the solution without any overshoot and the first estimate is
$$x_0=\frac{a}{a b+1}$$
Using for example $a=3$ and $b=2$, the iterates will be
$$\left(
\begin{array}{cc}
n & x_n \\
 0 & 0.428571 \\
 1 & 0.676719 \\
 2 & 0.740595 \\
 3 & 0.744102 \\
 4 & 0.744112
\end{array}
\right)$$
The exponential decreases so fast that tne next will be at almost a distance of $\pi$. Repeating the calculations
$$\left(
\begin{array}{cc}
n & x_n \\
 0 & 3.88571 \\
 1 & 2.96014 \\
 2 & 3.13834 \\
 3 & 3.13593
\end{array}
\right)$$
and again for the third
$$\left(
\begin{array}{cc}
n & x_n \\
 0 & 6.27752 \\
 1 & 6.28320
\end{array}
\right)$$
