Power series expansion involving Lambert-W function

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up vote
2
down vote

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Need to power series expand this term:

$ -frace2z Wleft(frac-2ze^2 right) left( log left( - frace2z W left( frac-2ze^2right) right) -1 right)$

I tried expanding this with Wolfram but it doesn't give me Higher Order Terms. Can anyone with Maple/Mathematica help out with this? I'm working on doing this by hand but it's gonna be tedious from the looks of it. Any suggestions welcome!!

sequences-and-series power-series taylor-expansion lambert-w

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edited Jul 26 at 19:22

Xoque55

2,65531328

asked Jul 26 at 19:10

anik faisal

154

add a comment | 

up vote
2
down vote

favorite

Need to power series expand this term:

$ -frace2z Wleft(frac-2ze^2 right) left( log left( - frace2z W left( frac-2ze^2right) right) -1 right)$

I tried expanding this with Wolfram but it doesn't give me Higher Order Terms. Can anyone with Maple/Mathematica help out with this? I'm working on doing this by hand but it's gonna be tedious from the looks of it. Any suggestions welcome!!

sequences-and-series power-series taylor-expansion lambert-w

share|cite|improve this question

edited Jul 26 at 19:22

Xoque55

2,65531328

asked Jul 26 at 19:10

anik faisal

154

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

Need to power series expand this term:

$ -frace2z Wleft(frac-2ze^2 right) left( log left( - frace2z W left( frac-2ze^2right) right) -1 right)$

I tried expanding this with Wolfram but it doesn't give me Higher Order Terms. Can anyone with Maple/Mathematica help out with this? I'm working on doing this by hand but it's gonna be tedious from the looks of it. Any suggestions welcome!!

sequences-and-series power-series taylor-expansion lambert-w

share|cite|improve this question

edited Jul 26 at 19:22

Xoque55

2,65531328

asked Jul 26 at 19:10

anik faisal

154

Need to power series expand this term:

$ -frace2z Wleft(frac-2ze^2 right) left( log left( - frace2z W left( frac-2ze^2right) right) -1 right)$

I tried expanding this with Wolfram but it doesn't give me Higher Order Terms. Can anyone with Maple/Mathematica help out with this? I'm working on doing this by hand but it's gonna be tedious from the looks of it. Any suggestions welcome!!

sequences-and-series power-series taylor-expansion lambert-w

share|cite|improve this question

edited Jul 26 at 19:22

Xoque55

2,65531328

asked Jul 26 at 19:10

anik faisal

154

share|cite|improve this question

share|cite|improve this question

edited Jul 26 at 19:22

Xoque55

2,65531328

edited Jul 26 at 19:22

Xoque55

2,65531328

edited Jul 26 at 19:22

Xoque55

2,65531328

2,65531328

asked Jul 26 at 19:10

anik faisal

154

asked Jul 26 at 19:10

anik faisal

154

asked Jul 26 at 19:10

anik faisal

154

154

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
0
down vote



accepted

Use the MATHEMATICA commands

f = -E/(2 z) ProductLog[-2 z/E^2] (Log [-E/(2 z) ProductLog[-2 z/E^2]] - 1)
Series[f, z, 1, 10]//N

and you will obtain

$$
f(z)=-0.880194-0.22445 (z-1.)-0.153651 (z-1.)^2-0.164248 (z-1.)^3-0.224172 (z-1.)^4-0.355448 (z-1.)^5-0.622729 (z-1.)^6-1.17077 (z-1.)^7-2.31929
(z-1.)^8-4.78304 (z-1.)^9-10.1829 (z-1.)^10+Oleft((z-1.)^11right)
$$

or if you prefer the lenghty form (only four therms)

$$
f(z) = frac12 e Wleft(-frac2e^2right) left(Wleft(-frac2e^2right)+2right)-frac12 left(e Wleft(-frac2e^2right)^2right)
(z-1)+frace Wleft(-frac2e^2right)^3 (z-1)^22 Wleft(-frac2e^2right)+2-fracleft(e Wleft(-frac2e^2right)^4 left(3
Wleft(-frac2e^2right)+4right)right) (z-1)^36 left(Wleft(-frac2e^2right)+1right)^3+frace Wleft(-frac2e^2right)^5
left(2 Wleft(-frac2e^2right) left(6 Wleft(-frac2e^2right)+17right)+25right) (z-1)^424
left(Wleft(-frac2e^2right)+1right)^5+Oleft((z-1)^5right)
$$

etc.

You can also use the fact

$$
f(z) = y(z)(ln y(z) -1)
$$

with

$$
y(z) = -frace Wleft(-frac2 ze^2right)2 z
$$

and

$$
y_0 = y(1) = -frac12 e Wleft(-frac2e^2right)
$$

and then find the expansion for $y$. This can be done with MATHEMATICA or with bare hand.

share|cite|improve this answer

edited Jul 26 at 20:28

answered Jul 26 at 20:03

Cesareo

5,5912412

• 
  
  
  

  
  

  
  
  
  
  
  

  
  thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
  – anik faisal
  Jul 26 at 20:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
  – Cesareo
  Jul 26 at 20:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  this is so very helpful!! thanks again
  – anik faisal
  Jul 26 at 20:21
  

  
  
  
  

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote



accepted

Use the MATHEMATICA commands

f = -E/(2 z) ProductLog[-2 z/E^2] (Log [-E/(2 z) ProductLog[-2 z/E^2]] - 1)
Series[f, z, 1, 10]//N

and you will obtain

$$
f(z)=-0.880194-0.22445 (z-1.)-0.153651 (z-1.)^2-0.164248 (z-1.)^3-0.224172 (z-1.)^4-0.355448 (z-1.)^5-0.622729 (z-1.)^6-1.17077 (z-1.)^7-2.31929
(z-1.)^8-4.78304 (z-1.)^9-10.1829 (z-1.)^10+Oleft((z-1.)^11right)
$$

or if you prefer the lenghty form (only four therms)

$$
f(z) = frac12 e Wleft(-frac2e^2right) left(Wleft(-frac2e^2right)+2right)-frac12 left(e Wleft(-frac2e^2right)^2right)
(z-1)+frace Wleft(-frac2e^2right)^3 (z-1)^22 Wleft(-frac2e^2right)+2-fracleft(e Wleft(-frac2e^2right)^4 left(3
Wleft(-frac2e^2right)+4right)right) (z-1)^36 left(Wleft(-frac2e^2right)+1right)^3+frace Wleft(-frac2e^2right)^5
left(2 Wleft(-frac2e^2right) left(6 Wleft(-frac2e^2right)+17right)+25right) (z-1)^424
left(Wleft(-frac2e^2right)+1right)^5+Oleft((z-1)^5right)
$$

etc.

You can also use the fact

$$
f(z) = y(z)(ln y(z) -1)
$$

with

$$
y(z) = -frace Wleft(-frac2 ze^2right)2 z
$$

and

$$
y_0 = y(1) = -frac12 e Wleft(-frac2e^2right)
$$

and then find the expansion for $y$. This can be done with MATHEMATICA or with bare hand.

share|cite|improve this answer

edited Jul 26 at 20:28

answered Jul 26 at 20:03

Cesareo

5,5912412

• 
  
  
  

  
  

  
  
  
  
  
  

  
  thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
  – anik faisal
  Jul 26 at 20:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
  – Cesareo
  Jul 26 at 20:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  this is so very helpful!! thanks again
  – anik faisal
  Jul 26 at 20:21
  

  
  
  
  

add a comment | 

up vote
0
down vote



accepted

Use the MATHEMATICA commands

f = -E/(2 z) ProductLog[-2 z/E^2] (Log [-E/(2 z) ProductLog[-2 z/E^2]] - 1)
Series[f, z, 1, 10]//N

and you will obtain

$$
f(z)=-0.880194-0.22445 (z-1.)-0.153651 (z-1.)^2-0.164248 (z-1.)^3-0.224172 (z-1.)^4-0.355448 (z-1.)^5-0.622729 (z-1.)^6-1.17077 (z-1.)^7-2.31929
(z-1.)^8-4.78304 (z-1.)^9-10.1829 (z-1.)^10+Oleft((z-1.)^11right)
$$

or if you prefer the lenghty form (only four therms)

$$
f(z) = frac12 e Wleft(-frac2e^2right) left(Wleft(-frac2e^2right)+2right)-frac12 left(e Wleft(-frac2e^2right)^2right)
(z-1)+frace Wleft(-frac2e^2right)^3 (z-1)^22 Wleft(-frac2e^2right)+2-fracleft(e Wleft(-frac2e^2right)^4 left(3
Wleft(-frac2e^2right)+4right)right) (z-1)^36 left(Wleft(-frac2e^2right)+1right)^3+frace Wleft(-frac2e^2right)^5
left(2 Wleft(-frac2e^2right) left(6 Wleft(-frac2e^2right)+17right)+25right) (z-1)^424
left(Wleft(-frac2e^2right)+1right)^5+Oleft((z-1)^5right)
$$

etc.

You can also use the fact

$$
f(z) = y(z)(ln y(z) -1)
$$

with

$$
y(z) = -frace Wleft(-frac2 ze^2right)2 z
$$

and

$$
y_0 = y(1) = -frac12 e Wleft(-frac2e^2right)
$$

and then find the expansion for $y$. This can be done with MATHEMATICA or with bare hand.

share|cite|improve this answer

edited Jul 26 at 20:28

answered Jul 26 at 20:03

Cesareo

5,5912412

• 
  
  
  

  
  

  
  
  
  
  
  

  
  thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
  – anik faisal
  Jul 26 at 20:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
  – Cesareo
  Jul 26 at 20:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  this is so very helpful!! thanks again
  – anik faisal
  Jul 26 at 20:21
  

  
  
  
  

add a comment | 

up vote
0
down vote



accepted

up vote
0
down vote



accepted

Use the MATHEMATICA commands

f = -E/(2 z) ProductLog[-2 z/E^2] (Log [-E/(2 z) ProductLog[-2 z/E^2]] - 1)
Series[f, z, 1, 10]//N

and you will obtain

$$
f(z)=-0.880194-0.22445 (z-1.)-0.153651 (z-1.)^2-0.164248 (z-1.)^3-0.224172 (z-1.)^4-0.355448 (z-1.)^5-0.622729 (z-1.)^6-1.17077 (z-1.)^7-2.31929
(z-1.)^8-4.78304 (z-1.)^9-10.1829 (z-1.)^10+Oleft((z-1.)^11right)
$$

or if you prefer the lenghty form (only four therms)

$$
f(z) = frac12 e Wleft(-frac2e^2right) left(Wleft(-frac2e^2right)+2right)-frac12 left(e Wleft(-frac2e^2right)^2right)
(z-1)+frace Wleft(-frac2e^2right)^3 (z-1)^22 Wleft(-frac2e^2right)+2-fracleft(e Wleft(-frac2e^2right)^4 left(3
Wleft(-frac2e^2right)+4right)right) (z-1)^36 left(Wleft(-frac2e^2right)+1right)^3+frace Wleft(-frac2e^2right)^5
left(2 Wleft(-frac2e^2right) left(6 Wleft(-frac2e^2right)+17right)+25right) (z-1)^424
left(Wleft(-frac2e^2right)+1right)^5+Oleft((z-1)^5right)
$$

etc.

You can also use the fact

$$
f(z) = y(z)(ln y(z) -1)
$$

with

$$
y(z) = -frace Wleft(-frac2 ze^2right)2 z
$$

and

$$
y_0 = y(1) = -frac12 e Wleft(-frac2e^2right)
$$

and then find the expansion for $y$. This can be done with MATHEMATICA or with bare hand.

share|cite|improve this answer

edited Jul 26 at 20:28

answered Jul 26 at 20:03

Cesareo

5,5912412

Use the MATHEMATICA commands

f = -E/(2 z) ProductLog[-2 z/E^2] (Log [-E/(2 z) ProductLog[-2 z/E^2]] - 1)
Series[f, z, 1, 10]//N

and you will obtain

$$
f(z)=-0.880194-0.22445 (z-1.)-0.153651 (z-1.)^2-0.164248 (z-1.)^3-0.224172 (z-1.)^4-0.355448 (z-1.)^5-0.622729 (z-1.)^6-1.17077 (z-1.)^7-2.31929
(z-1.)^8-4.78304 (z-1.)^9-10.1829 (z-1.)^10+Oleft((z-1.)^11right)
$$

or if you prefer the lenghty form (only four therms)

$$
f(z) = frac12 e Wleft(-frac2e^2right) left(Wleft(-frac2e^2right)+2right)-frac12 left(e Wleft(-frac2e^2right)^2right)
(z-1)+frace Wleft(-frac2e^2right)^3 (z-1)^22 Wleft(-frac2e^2right)+2-fracleft(e Wleft(-frac2e^2right)^4 left(3
Wleft(-frac2e^2right)+4right)right) (z-1)^36 left(Wleft(-frac2e^2right)+1right)^3+frace Wleft(-frac2e^2right)^5
left(2 Wleft(-frac2e^2right) left(6 Wleft(-frac2e^2right)+17right)+25right) (z-1)^424
left(Wleft(-frac2e^2right)+1right)^5+Oleft((z-1)^5right)
$$

etc.

You can also use the fact

$$
f(z) = y(z)(ln y(z) -1)
$$

with

$$
y(z) = -frace Wleft(-frac2 ze^2right)2 z
$$

and

$$
y_0 = y(1) = -frac12 e Wleft(-frac2e^2right)
$$

and then find the expansion for $y$. This can be done with MATHEMATICA or with bare hand.

share|cite|improve this answer

edited Jul 26 at 20:28

answered Jul 26 at 20:03

Cesareo

5,5912412

share|cite|improve this answer

share|cite|improve this answer

edited Jul 26 at 20:28

edited Jul 26 at 20:28

edited Jul 26 at 20:28

answered Jul 26 at 20:03

Cesareo

5,5912412

answered Jul 26 at 20:03

Cesareo

5,5912412

answered Jul 26 at 20:03

Cesareo

5,5912412

5,5912412

• 
  
  
  

  
  

  
  
  
  
  
  

  
  thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
  – anik faisal
  Jul 26 at 20:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
  – Cesareo
  Jul 26 at 20:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  this is so very helpful!! thanks again
  – anik faisal
  Jul 26 at 20:21
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
  – anik faisal
  Jul 26 at 20:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
  – Cesareo
  Jul 26 at 20:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  this is so very helpful!! thanks again
  – anik faisal
  Jul 26 at 20:21
  

  
  
  
  

thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
– anik faisal
Jul 26 at 20:07

thanks so much! this expansion is about z=0, I also need to expand it about z=1. Could you help out with that?
– anik faisal
Jul 26 at 20:07

@anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
– Cesareo
Jul 26 at 20:15

@anikfaisal OK. Now the expansion near $z=1$ To avoid length formulations, use the command //N
– Cesareo
Jul 26 at 20:15

this is so very helpful!! thanks again
– anik faisal
Jul 26 at 20:21

this is so very helpful!! thanks again
– anik faisal
Jul 26 at 20:21

add a comment | 

 

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