added possible generalization section

1 files changed, 15 insertions, 2 deletions

diff --git a/execsumm/document.tex b/execsumm/document.tex
index 64f12d7..85d2e87 100644
--- a/execsumm/document.tex
+++ b/execsumm/document.tex
@@ -65,9 +65,9 @@ $\displaystyle z = y{C_2R_1\over r+C_1R_1}.$
 This corresponds to a solution of the form $f = Ce^{\lambda t}v,\cdots$
 where $C\in {\bf C},$ $f\in {\bf R}\to{\bf R}$. If the non-trivial
 eigenvectors are complex, %% TRY TO PROVE THIS!!
-\def\re{{\rm Re}}\def\im{{\rm Im}}
 their exponential solutions form, in the reals,
-$g = C_1\cos(\re(\lambda)t)\re(v) + C_2\sin(\re(\lambda)t)\im(v).$ %% DOUBLE CHECK.
+\def\re{{\rm Re}}\def\im{{\rm Im}}
+$g = C_1\cos(\re(\lambda)t)\re(v) + C_2\sin(\re(\lambda)t)\im(v).$
 
 \section{Nonhomogeneous System}
 
@@ -79,4 +79,17 @@ undetermined coefficients, which may be attempted, but variation of
 parameters will be used so that a general form for both systems with
 both complex and real eigenvalues may be found. 
 
+\section{Possible Generalization}
+
+This solution is general to any formulation of the original problem,
+but gain may look different for a square or triangular wave, for
+example. It is expected that these would exhibit similar behavior
+to the sine wave because, because they would input similar amounts of
+energy on a similar time scale, but the sensitivity to waveform type
+could be investigated in the same way that this paper did, with possible
+usage of the Laplace transform to handle discontinuities, but because
+there is a general form for a sine wave and the output is proportional
+to the input, a Fourier transform could be used to either approximate
+or analytically obtain a solution for these types of waves.
+
 \bye