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An exponentially modified Gaussian distribution is often encountered in modelling chromatographic peaks[ 1] [ 2] [ 3] , [ 4] [ 5]
f
(
x
)
=
y
0
+
A
t
0
e
x
p
(
1
2
(
w
t
0
)
2
−
x
−
x
c
t
0
)
∫
−
∞
z
1
2
e
x
p
(
−
y
2
z
)
d
y
{\displaystyle f(x)=y_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\int _{-\infty }^{z}{\frac {1}{\sqrt {2}}}exp\left({\frac {-y^{2}}{z}}\right)dy}
or
f
(
x
)
=
y
0
+
A
t
0
e
x
p
(
1
2
(
w
t
0
)
2
−
x
−
x
c
t
0
)
(
1
2
+
1
2
e
r
f
(
z
2
)
)
{\displaystyle f(x)=y_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\left({\frac {1}{2}}+{\frac {1}{2}}erf\left({\frac {z}{\sqrt {2}}}\right)\right)}
y
0
{\displaystyle y_{0}}
A
{\displaystyle A}
x
c
{\displaystyle x_{c}}
w
{\displaystyle w}
t
0
{\displaystyle t_{0}}
t
0
>
0
{\displaystyle t_{0}>0}
z
=
x
−
x
c
w
−
w
t
0
{\displaystyle z={\frac {x-x_{c}}{w}}-{\frac {w}{t_{0}}}}
e
r
f
(
z
2
)
{\displaystyle erf\left({\frac {z}{\sqrt {2}}}\right)}
error function of
z
2
{\displaystyle {\frac {z}{\sqrt {2}}}}
References
