Peak Functions¶
Full Mathematical Format¶
Non-Linear Gaussian Functions¶
2001 z=aexp(-0.5*(((x-b)/c)2+((y-d)/e)2))
2002 z=a+bexp(-0.5*(((x-c)/d)2+((y-e)/f)2))
2003 z=aexp(-0.5*(((x-b)/c)2+((y-d)/c)2))
2004 z=a+bexp(-0.5*(((x-c)/d)2+((y-e)/d)2))
2005 z=aexp(-0.5*((x-b)/c)2)+dexp(-0.5*((y-e)/f)2)
2006 z=a+bexp(-0.5*((x-c)/d)2)+eexp(-0.5*((y-f)/g)2)
2007 z=a+by+cexp(-0.5*((x-d)/e)2)
2008 z=a+bx+cexp(-0.5*((y-d)/e)2)
2009 z=a+by+cy2+dexp(-0.5*((x-e)/f)2)
2010 z=a+bx+cx2+dexp(-0.5*((y-e)/f)2)
2011 z=aexp(-0.5*((x-b)/c)2)+dexp(-0.5*((y-e)/f)2)+gexp(-0.5*(((x-b)/c)2+((y-e)/f)2))
2012 z=a+bexp(-0.5*((x-c)/d)2)+eexp(-0.5*((y-f)/g)2)+hexp(-0.5*(((x-c)/d)2+((y-f)/g)2))
Non-Linear Lorentzian Functions¶
2013 z=a/((1+((x-b)/c)2)*(1+((y-d)/e)2))
2014 z=a+b/((1+((x-c)/d)2)*(1+((y-e)/f)2))
2015 z=a/((1+((x-b)/c)2)*(1+((y-d)/c)2))
2016 z=a+b/((1+((x-c)/d)2)*(1+((y-e)/d)2))
2017 z=a/(1+((x-b)/c)2)+d/(1+((y-e)/f)2)
2018 z=a+b/(1+((x-c)/d)2)+e/(1+((y-f)/g)2)
2019 z=a+by+c/(1+((x-d)/e)2)
2020 z=a+bx+c/(1+((y-d)/e)2)
2021 z=a+by+cy2+d/(1+((x-e)/f)2)
2022 z=a+bx+cx2+d/(1+((y-e)/f)2)
2023 z=a/(1+((x-b)/c)2)+d/(1+((y-e)/f)2)+g/((1+((x-b)/c)2)*(1+((y-e)/f)2))
2024 z=a+b/(1+((x-c)/d)2)+e/(1+((y-f)/g)2)+h/((1+((x-c)/d)2)*(1+((y-f)/g)2))
Non-Linear Log-Normal Functions¶
2025 z=aexp(-0.5*((ln(x/b)/c)2+(ln(y/d)/e)2))
2026 z=a+bexp(-0.5*((ln(x/c)/d)2+(ln(y/e)/f)2))
2027 z=aexp(-0.5*((ln(x/b)/c)2+(ln(y/d)/c)2))
2028 z=a+bexp(-0.5*((ln(x/c)/d)2+(ln(y/e)/d)2))
2029 z=aexp(-0.5*(ln(x/b)/c)2)+dexp(-0.5*(ln(y/e)/f)2)
2030 z=a+bexp(-0.5*(ln(x/c)/d)2)+eexp(-0.5*(ln(y/f)/g)2)
2031 z=a+by+cexp(-0.5*(ln(x/d)/e)2)
2032 z=a+bx+cexp(-0.5*(ln(y/d)/e)2)
2033 z=a+by+cy2+dexp(-0.5*(ln(x/e)/f)2)
2034 z=a+bx+cx2+dexp(-0.5*(ln(y/e)/f)2)
2035 z=aexp(-0.5*(ln(x/b)/c)2)+dexp(-0.5*(ln(y/e)/f)2)+gexp(-0.5*((ln(x/b)/c)2+(ln(y/e)/f)2))
2036 z=a+bexp(-0.5*(ln(x/c)/d)2)+eexp(-0.5*(ln(y/f)/g)2)+hexp(-0.5*((ln(x/c)/d)2+(ln(y/f)/g)2))
Non-Linear Logistic Peak Functions¶
2037 z=16aexp(-((x-b)/c)-((y-d)/e))/((1+exp(-((x-b)/c)))2*(1+exp(-((y-d)/e)))2)
2038 z=a+16bexp(-((x-c)/d)-((y-e)/f))/((1+exp(-((x-c)/d)))2*(1+exp(-((y-e)/f)))2)
2039 z=16aexp(-((x-b)/c)-((y-d)/c))/((1+exp(-((x-b)/c)))2*(1+exp(-((y-d)/c)))2)
2040 z=a+16bexp(-((x-c)/d)-((y-e)/d))/((1+exp(-((x-c)/d)))2*(1+exp(-((y-e)/d)))2)
2041 z=4aexp(-((x-b)/c))/((1+exp(-((x-b)/c)))2)+4dexp(-((y-e)/f))/((1+exp(-((y-e)/f)))2)
2042 z=a+4bexp(-((x-c)/d))/((1+exp(-((x-c)/d)))2)+4eexp(-((y-f)/g))/((1+exp(-((y-f)/g)))2)
2043 z=a+by+4cexp(-((x-d)/e))/((1+exp(-((x-d)/e)))2)
2044 z=a+bx+4cexp(-((y-d)/e))/((1+exp(-((y-d)/e)))2)
2045 z=a+by+cy2+4dexp(-((x-e)/f))/((1+exp(-((x-e)/f)))2)
2046 z=a+bx+cx2+4dexp(-((y-e)/f))/((1+exp(-((y-e)/f)))2)
2047 z=4aexp(-((x-b)/c))/((1+exp(-((x-b)/c)))2)+4dexp(-((y-e)/f))/((1+exp(-((y-e)/f)))2)+16gexp(-((x-b)/c)-((y-e)/f))/((1+exp(-((x-b)/c)))2*(1+exp(-((y-e)/f)))2)
2048 z=a+4bexp(-((x-c)/d))/((1+exp(-((x-c)/d)))2)+4eexp(-((y-f)/g))/((1+exp(-((y-f)/g)))2)+16hexp(-((x-c)/d)-((y-f)/g))/((1+exp(-((x-c)/d)))2*(1+exp(-((y-f)/g)))2)
Non-Linear Extreme Value Functions¶
2049 z=aexp(-exp(-(x-b)/c)-(x-b)/c+1)*exp(-exp(-(y-d)/e)-(y-d)/e+1)
2050 z=a+bexp(-exp(-(x-c)/d)-(x-c)/d+1)*exp(-exp(-(y-e)/f)-(y-e)/f+1)
2051 z=aexp(-exp(-(x-b)/c)-(x-b)/c+1)*exp(-exp(-(y-d)/c)-(y-d)/c+1)
2052 z=a+bexp(-exp(-(x-c)/d)-(x-c)/d+1)*exp(-exp(-(y-e)/d)-(y-e)/d+1)
2053 z=aexp(-exp(-(x-b)/c)-(x-b)/c+1)+dexp(-exp(-(y-e)/f)-(y-e)/f+1)
2054 z=a+bexp(-exp(-(x-c)/d)-(x-c)/d+1)+eexp(-exp(-(y-f)/g)-(y-f)/g+1)
2055 z=a+by+cexp(-exp(-(x-d)/e)-(x-d)/e+1)
2056 z=a+bx+cexp(-exp(-(y-d)/e)-(y-d)/e+1)
2057 z=a+by+cy2+dexp(-exp(-(x-e)/f)-(x-e)/f+1)
2058 z=a+bx+cx2+dexp(-exp(-(y-e)/f)-(y-e)/f+1)
2059 z=aexp(-exp(-(x-b)/c)-(x-b)/c+1)+dexp(-exp(-(y-e)/f)-(y-e)/f+1)+gexp(-exp(-(x-b)/c)-(x-b)/c+1)*exp(-exp(-(y-e)/f)-(y-e)/f+1)
2060 z=a+bexp(-exp(-(x-c)/d)-(x-c)/d+1)+eexp(-exp(-(y-f)/g)-(y-f)/g+1)+hexp(-exp(-(x-c)/d)-(x-c)/d+1)*exp(-exp(-(y-f)/g)-(y-f)/g+1)
Non-Linear Gaussian/Log-Normal Functions¶
2061 z=aexp(-0.5*(((x-b)/c)2+(ln(y/d)/e)2))
2062 z=aexp(-0.5*((ln(x/b)/c)2+((y-d)/e)2))
2063 z=a+bexp(-0.5*(((x-c)/d)2+(ln(y/e)/f)2))
2064 z=a+bexp(-0.5*((ln(x/c)/d)2+((y-e)/f)2))
2065 z=aexp(-0.5*(((x-b)/c)2))+dexp(-0.5*((ln(y/e)/f)2))
2066 z=aexp(-0.5*((ln(x/b)/c)2))+dexp(-0.5*(((y-e)/f)2))
2067 z=a+bexp(-0.5*(((x-c)/d)2))+eexp(-0.5*((ln(y/f)/g)2))
2068 z=a+bexp(-0.5*((ln(x/c)/d)2))+eexp(-0.5*(((y-f)/g)2))
2069 z=aexp(-0.5*(((x-b)/c)2))+dexp(-0.5*((ln(y/e)/f)2))+gexp(-0.5*(((x-b)/c)2+(ln(y/e)/f)2))
2070 z=aexp(-0.5*((ln(x/b)/c)2))+dexp(-0.5*(((y-e)/f)2))+gexp(-0.5*((ln(x/b)/c)2+((y-e)/f)2))
2071 z=a+bexp(-0.5*(((x-c)/d)2))+eexp(-0.5*((ln(y/f)/g)2))+hexp(-0.5*(((x-c)/d)2+(ln(y/f)/g)2))
2072 z=a+bexp(-0.5*((ln(x/c)/d)2))+eexp(-0.5*(((y-f)/g)2))+hexp(-0.5*((ln(x/c)/d)2+((y-f)/g)2))
Notational Format¶
Non-Linear Gaussian Functions¶
2001 z=GAUSSX(a,b,c)*GAUSSY(1,d,e)
2002 z=a+GAUSSX(b,c,d)*GAUSSY(1,e,f)
2003 z=GAUSSX(a,b,c)*GAUSSY(1,d,c)
2004 z=a+GAUSSX(b,c,d)*GAUSSY(1,e,d)
2005 z=GAUSSX(a,b,c)+GAUSSY(d,e,f)
2006 z=a+GAUSSX(b,c,d)+GAUSSY(e,f,g)
2007 z=a+by+GAUSSX(c,d,e)
2008 z=a+bx+GAUSSY(c,d,e)
2009 z=a+by+cy2+GAUSSX(d,e,f)
2010 z=a+bx+cx2+GAUSSY(d,e,f)
2011 z=GAUSSX(a,b,c)+GAUSSY(d,e,f)+GAUSSX(g,b,c)*GAUSSY(1,e,f)
2012 z=a+GAUSSX(b,c,d)+GAUSSY(e,f,g)+GAUSSX(h,c,d)*GAUSSY(1,f,g)
Non-Linear Lorentzian Functions¶
2013 z=LORX(a,b,c)*LORY(1,d,e)
2014 z=a+LORX(b,c,d)*LORY(1,e,f)
2015 z=LORX(a,b,c)*LORY(1,d,c)
2016 z=a+LORX(b,c,d)*LORY(1,e,d)
2017 z=LORX(a,b,c)+LORY(d,e,f)
2018 z=a+LORX(b,c,d)+LORY(e,f,g)
2019 z=a+by+LORX(c,d,e)
2020 z=a+bx+LORY(c,d,e)
2021 z=a+by+cy2+LORX(d,e,f)
2022 z=a+bx+cx2+LORY(d,e,f)
2023 z=LORX(a,b,c)+LORY(d,e,f)+LORX(g,b,c)*LORY(1,e,f)
2024 z=a+LORX(b,c,d)+LORY(e,f,g)+LORX(h,c,d)*LORY(1,f,g)
Non-Linear Log Normal Functions¶
2025 z=LOGNORMX(a,b,c)*LOGNORMY(1,d,e)
2026 z=a+LOGNORMX(b,c,d)*LOGNORMY(1,e,f)
2027 z=LOGNORMX(a,b,c)*LOGNORMY(1,d,c)
2028 z=a+LOGNORMX(b,c,d)*LOGNORMY(1,e,d)
2029 z=LOGNORMX(a,b,c)+LOGNORMY(d,e,f)
2030 z=a+LOGNORMX(b,c,d)+LOGNORMY(e,f,g)
2031 z=a+by+LOGNORMX(c,d,e)
2032 z=a+bx+LOGNORMY(c,d,e)
2033 z=a+by+cy2+LOGNORMX(d,e,f)
2034 z=a+bx+cx2+LOGNORMY(d,e,f)
2035 z=LOGNORMX(a,b,c)+LOGNORMY(d,e,f)+LOGNORMX(g,b,c)*LOGNORMY(1,e,f)
2036 z=a+LOGNORMX(b,c,d)+LOGNORMY(e,f,g)+LOGNORMX(h,c,d)*LOGNORMY(1,f,g)
Non-Linear Logistic Peak Functions¶
2037 z=LOGISTICX(a,b,c)*LOGISTICY(1,d,e)
2038 z=a+LOGISTICX(b,c,d)*LOGISTICY(1,e,f)
2039 z=LOGISTICX(a,b,c)*LOGISTICY(1,d,c)
2040 z=a+LOGISTICX(b,c,d)*LOGISTICY(1,e,d)
2041 z=LOGISTICX(a,b,c)+LOGISTICY(d,e,f)
2042 z=a+LOGISTICX(b,c,d)+LOGISTICY(e,f,g)
2043 z=a+by+LOGISTICX(c,d,e)
2044 z=a+bx+LOGISTICY(c,d,e)
2045 z=a+by+cy2+LOGISTICX(d,e,f)
2046 z=a+bx+cx2+LOGISTICY(d,e,f)
2047 z=LOGISTICX(a,b,c)+LOGISTICY(d,e,f)+LOGISTICX(g,b,c)*LOGISTICY(1,e,f)
2048 z=a+LOGISTICX(b,c,d)+LOGISTICY(e,f,g)+LOGISTICX(h,c,d)*LOGISTICY(1,f,g)
Non-Linear Extreme Value Functions¶
2049 z=EXTRVALX(a,b,c)*EXTRVALY(1,d,e)
2050 z=a+EXTRVALX(b,c,d)*EXTRVALY(1,e,f)
2051 z=EXTRVALX(a,b,c)*EXTRVALY(1,d,c)
2052 z=a+EXTRVALX(b,c,d)*EXTRVALY(1,e,d)
2053 z=EXTRVALX(a,b,c)+EXTRVALY(d,e,f)
2054 z=a+EXTRVALX(b,c,d)+EXTRVALY(e,f,g)
2055 z=a+by+EXTRVALX(c,d,e)
2056 z=a+bx+EXTRVALY(c,d,e)
2057 z=a+by+cy2+EXTRVALX(d,e,f)
2058 z=a+bx+cx2+EXTRVALY(d,e,f)
2059 z=EXTRVALX(a,b,c)+EXTRVALY(d,e,f)+EXTRVALX(g,b,c)*EXTRVALY(1,e,f)
2060 z=a+EXTRVALX(b,c,d)+EXTRVALY(e,f,g)+EXTRVALX(h,c,d)*EXTRVALY(1,f,g)
Non-Linear Gaussian/Log-Normal Functions¶
2061 z=GAUSSX(a,b,c)*LOGNORMY(1,d,e)
2062 z=LOGNORMX(a,b,c)*GAUSSY(1,d,e)
2063 z=a+GAUSSX(b,c,d)*LOGNORMY(1,e,f)
2064 z=a+LOGNORMX(b,c,d)*GAUSSY(1,e,f)
2065 z=GAUSSX(a,b,c)+LOGNORMY(d,e,f)
2066 z=LOGNORMX(a,b,c)+GAUSSY(d,e,f)
2067 z=a+GAUSSX(b,c,d)+LOGNORMY(e,f,g)
2068 z=a+LOGNORMX(b,c,d)+GAUSSY(e,f,g)
2069 z=GAUSSX(a,b,c)+LOGNORMY(d,e,f)+GAUSSX(g,b,c)*LOGNORMY(1,e,f)
2070 z=LOGNORMX(a,b,c)+GAUSSY(d,e,f)+LOGNORMX(g,b,c)*GAUSSY(1,e,f)
2071 z=a+GAUSSX(b,c,d)+LOGNORMY(e,f,g)+GAUSSX(h,c,d)*LOGNORMY(1,f,g)
2072 z=a+LOGNORMX(b,c,d)+GAUSSY(e,f,g)+LOGNORMX(h,c,d)*GAUSSY(1,f,g)