Asymmetric bell
Asymmetric bell (or «Asymmetric bell function») is kind of functions usable for approximation of shapes of simple peaks, with hope that the precision of the approximation is sufficient fo work with some special function (preferably an elementary function) instead of complicated numerical solution or experimental data.
Use of the approximations (with minimal number of parameters), while its precision is still sufficient, corresponds to the 6th of the TORI axioms, as the elementary function is expected to be easier, simpler, than the numeric solution or the experimental data.
Definition
Asymmetric bell is any 4-parametric real-holomorphic function \(F\) with the following properties:
1. for some positive \(A\), \(F(z)\) is holomorphic in some strip \( |\Im(z)| < A \)
2. At the real axis, \(F\) is positive,
\( F(x)>0 \) for real \( x \)
3. \(F\) decays to zero at \(\pm \infty\):
\( \displaystyle \lim_{x\rightarrow \pm \infty}F(x)=0 \)
4. Function \( F \) has single maximum with negative second derivative at this maximum:
there exist \( x_0 \) such that
\( F(x) < F(x_0) \) for any \(x \ne x_0 \)
5. Function is monotonous at each of the two halflines,
\( F^{\prime}(x) >0 \) for \(x < x_0 \) and \( F^{\prime}(x) <0 \) for \(x > x_0 \)
6. Function \( F \) has two and only two saddle points \(x_1\) and \(x_2 \) such that
\(x_1 < x_0 < x_2\)
\( F^{\prime\prime}(x_1)=F^{\prime\prime}(x_2)=0 \)
\( F^{\prime\prime\prime}(x_1)<0 \) and \(F^{\prime\prime\prime}(x_2)>0 \)
Of course, with conditions above, \( F^{\prime}(x_1)>0 \) and \(F^{\prime}(x_2)<0 \)
7. At some 3-parametric set of values of the parameters, the \(F\) is symmetric bell function,
id est,
for the \(x_0\), the symmetry holds:
\( F(x_0+x)=F(x_0-x) \)
Example
\( \displaystyle F(x) = \frac {1}{ A\ \exp(px) + B\ \exp(-qx) } \)
with positive real parameters \(A,B,p,q\).
The maximum \(x_0 \) is solution of equation
\( A p \exp(px_0) = B q \exp(-qx_0) \)
\( \exp((p+q)x_0) = \frac{Bq}{Ap} \)
\(\displaystyle x_0 = \log_{p+q} \left( \frac{Bq}{Ap} \right) \)
At \(p=q\), function \(F\) becomes symmetric bell function.
Warning
The bell function should not be confused with a Bellman, even if the Bellman presents to perform some important function.
References
http://www.multichrom.ru/Docs/approxim.pdf Аппроксимация асимметричных пиков