Effective Wavelength
We desire to determine an effective wavelength, λ0, for a filter such that
F  F0 W ,

where Fλ is the monochromatic flux, F   F T d is the total (integrated) flux,
0

W   T d is the filter width and Tλ is the filter transmission function. To find λ0 we
0
expand Fλ in a Taylor series about λ0 :
 dF ( ) 
F ( )  F ( 0 )  
 (   0 )  higher order term s in (   0 ).
 d  0
If the higher order terms are negligibly small, then


 dF ( )  
 dF ( )  
F   F T d  F ( 0 )  T d 
(



)
T
d


F
(

)
W



  (   0 )T d .
0

0

0
0
 d  0 0
 d  0 0
We note that if


0
(  0 )T d  0, then F  F0 W .
This will be true if
0


0
0
 T d  

 T d

T d
W
.
0
So F  F ( 0 )W , where the effective wavelength, λ0, is determined by  0



0
T d
W
.