Changes between Version 14 and Version 15 of SpectraObject

Timestamp:

01/13/15 10:51:22 (10 years ago)

Author:

Jonathan

Comment:

—

SpectraObject

@@ -95 +95 @@
 [[latex($F_k=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} k x} f_{x}$)]]
 
-where [[latex($k$)]] is any integer.  Now [[latex($F_k$)]] is periodic in [[latex($k$)]] since
+where [[latex($k$)]] is any integer.  
+
+Now [[latex($F_k$)]] is periodic in [[latex($k$)]] since
 
 [[latex($F_{k+N_x}=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} \left(k+N_x\right) x} f_{x} = \displaystyle \sum_{x=1}^{N_x} e^{2 \pi i x} \times e^{  \frac{2 \pi i}{N_x} k x} f_{x}=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} kx} f_{x}  =F_k$)]]
+
+We can visually see this by plotting 
+
+[[Image(Screen Shot 2015-01-06 at 3.38.19 PM.png​,width=600)]]
+
+
 
 so while we can calculate the transform for any k, only N of them will be unique.
@@ -122 +130 @@
 The real part of the discrete function for l = 2 and l = 10 are coincident! As are 3 and 9, 4 and 8, 5 and 7.  
 
-[[Image(Screen Shot 2015-01-06 at 3.38.19 PM.png​,width=600)]]
 
 What about the imaginary parts?