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Mathematics Algebra Calculus Wolfram Language

Calculate the sum of a series of lorentzain function?

Yu Li

Posted 7 years ago

Consider the following code: IN: = Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}] Out: = (\[Pi] Sinh[2 a \[Pi]])/(a (-Cos[2 \[Pi] x] + Cosh[2 a \[Pi]])) But if I input Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}] just by multiplying a factor a, the result is totally different. 1/2 I (PolyGamma[0, 1 - I a - x] - PolyGamma[0, 1 + I a - x] + PolyGamma[0, -I a + x] - PolyGamma[0, I a + x]) Apparently there is something wrong here... Could anybody help me?

Consider the following code:

IN: = Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}]


Out: = (\[Pi] Sinh[2 a \[Pi]])/(a (-Cos[2 \[Pi] x] + Cosh[2 a \[Pi]]))

But if I input

Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}]

just by multiplying a factor a, the result is totally different.

1/2 I (PolyGamma[0, 1 - I a - x] - PolyGamma[0, 1 + I a - x] + 
   PolyGamma[0, -I a + x] - PolyGamma[0, I a + x])

Apparently there is something wrong here... Could anybody help me?

POSTED BY: Yu Li

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Valeriu Ungureanu

Valeriu Ungureanu, Moldova State University

Posted 7 years ago

You can apply FullSimplify[] and the result will be more appropriate with your expectations: In[10]:= Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}] Out[10]= (\[Pi] Sinh[2 a \[Pi]])/(a (-Cos[2 \[Pi] x] + Cosh[2 a \[Pi]])) In[11]:= Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}] Out[11]= 1/2 (\[Pi] Coth[a \[Pi] - I \[Pi] x] + \[Pi] Coth[a \[Pi] + I \[Pi] x]) In[12]:= FullSimplify[1/2 I (PolyGamma[0, 1 - I a - x] - PolyGamma[0, 1 + I a - x] + PolyGamma[0, -I a + x] - PolyGamma[0, I a + x])] Out[12]= 1/2 \[Pi] (Coth[\[Pi] (a - I x)] + Coth[\[Pi] (a + I x)])

You can apply FullSimplify[] and the result will be more appropriate with your expectations:

In[10]:= Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}]
Out[10]= (\[Pi] Sinh[2 a \[Pi]])/(a (-Cos[2 \[Pi] x] + Cosh[2 a \[Pi]]))

In[11]:= Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}]
Out[11]= 1/2 (\[Pi] Coth[a \[Pi] - I \[Pi] x] + \[Pi] Coth[a \[Pi] + I \[Pi] x])

In[12]:= FullSimplify[1/2 I (PolyGamma[0, 1 - I a - x] - PolyGamma[0, 1 + I a - x] + PolyGamma[0, -I a + x] - PolyGamma[0, I a + x])]
Out[12]= 1/2 \[Pi] (Coth[\[Pi] (a - I x)] + Coth[\[Pi] (a + I x)])

POSTED BY: Valeriu Ungureanu

Yu Li

Posted 7 years ago

Thanks. FullSimplify did help. and if we set S1 = Sum[1/((x + n)^2 + a^2), {n, -Infinity, Infinity}]; S2 = Sum[a/((x + n)^2 + a^2), {n, -Infinity, Infinity}]; S1*a-S2 turn out to be zero.

POSTED BY: Yu Li

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