Why Mathematica does nothing with the expression Gamma[2 z]? [on hold]

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0
down vote

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Could someone tell me why Mathematica just returns the input with this expression:

Gamma[2z]

Gamma[2 z]

?

I expected this result:

$Γ(2z)=frac2^2z-1Γ(z)Γ(z+1/2)sqrtpi$

functions simplifying-expressions special-functions

share|improve this question

edited Sep 5 at 23:53

Carl Woll

56.2k272147

asked Sep 5 at 21:51

Isa

1276

put on hold as off-topic by AccidentalFourierTransform, m_goldberg, Daniel Lichtblau, Henrik Schumacher, José Antonio Díaz Navas Sep 7 at 11:39

• The question does not concern the technical computing software Mathematica by Wolfram Research. Please see the help center to find out about the topics that can be asked here.

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Why should it? The right hand side is much more complicated than the left one.
  – Henrik Schumacher
  Sep 5 at 21:52
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Yes, it is the $Gamma$-function. But how on earth should Mathematica tell that you want to have the expression Gamma[2 z] expanded in exactly this way? For example, it could also return Gamma[2 z - 1] (2 z - 1). Or any other identity involving the $Gamma$-function (there are probably really many identities at least as interesting as the one you gave). What I tried to explain to you: i) Computers cannot read your mind and ii) Mathematica just doesn't work this way.
  – Henrik Schumacher
  Sep 5 at 22:06
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It's not a command. Don't think of it that way. It's an expression. Mathematica is an expression rewriting language. If you want it to rewrite an expression in a particular way, you must tell it.
  – John Doty
  Sep 5 at 22:59
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I'm voting to close this question as off-topic because the issued raised is not really a problem; it is arises from the OP's misunderstanding of the result returned by Mathematica.
  – m_goldberg
  Sep 5 at 23:16
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  That rhs looks like gamma after she got run over by a reindeer...
  – Daniel Lichtblau
  Sep 6 at 4:18
  

  
  
  
  

 | 
show 8 more comments

up vote
0
down vote

favorite

1

Could someone tell me why Mathematica just returns the input with this expression:

Gamma[2z]

Gamma[2 z]

?

I expected this result:

$Γ(2z)=frac2^2z-1Γ(z)Γ(z+1/2)sqrtpi$

functions simplifying-expressions special-functions

share|improve this question

edited Sep 5 at 23:53

Carl Woll

56.2k272147

asked Sep 5 at 21:51

Isa

1276

put on hold as off-topic by AccidentalFourierTransform, m_goldberg, Daniel Lichtblau, Henrik Schumacher, José Antonio Díaz Navas Sep 7 at 11:39

• The question does not concern the technical computing software Mathematica by Wolfram Research. Please see the help center to find out about the topics that can be asked here.

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Why should it? The right hand side is much more complicated than the left one.
  – Henrik Schumacher
  Sep 5 at 21:52
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Yes, it is the $Gamma$-function. But how on earth should Mathematica tell that you want to have the expression Gamma[2 z] expanded in exactly this way? For example, it could also return Gamma[2 z - 1] (2 z - 1). Or any other identity involving the $Gamma$-function (there are probably really many identities at least as interesting as the one you gave). What I tried to explain to you: i) Computers cannot read your mind and ii) Mathematica just doesn't work this way.
  – Henrik Schumacher
  Sep 5 at 22:06
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It's not a command. Don't think of it that way. It's an expression. Mathematica is an expression rewriting language. If you want it to rewrite an expression in a particular way, you must tell it.
  – John Doty
  Sep 5 at 22:59
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I'm voting to close this question as off-topic because the issued raised is not really a problem; it is arises from the OP's misunderstanding of the result returned by Mathematica.
  – m_goldberg
  Sep 5 at 23:16
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  That rhs looks like gamma after she got run over by a reindeer...
  – Daniel Lichtblau
  Sep 6 at 4:18
  

  
  
  
  

 | 
show 8 more comments

up vote
0
down vote

favorite

1

up vote
0
down vote

favorite

1

1

Could someone tell me why Mathematica just returns the input with this expression:

Gamma[2z]

Gamma[2 z]

?

I expected this result:

$Γ(2z)=frac2^2z-1Γ(z)Γ(z+1/2)sqrtpi$

functions simplifying-expressions special-functions

share|improve this question

edited Sep 5 at 23:53

Carl Woll

56.2k272147

asked Sep 5 at 21:51

Isa

1276

Could someone tell me why Mathematica just returns the input with this expression:

Gamma[2z]

Gamma[2 z]

?

I expected this result:

$Γ(2z)=frac2^2z-1Γ(z)Γ(z+1/2)sqrtpi$

functions simplifying-expressions special-functions

share|improve this question

edited Sep 5 at 23:53

Carl Woll

56.2k272147

asked Sep 5 at 21:51

Isa

1276

share|improve this question

share|improve this question

edited Sep 5 at 23:53

Carl Woll

56.2k272147

edited Sep 5 at 23:53

Carl Woll

56.2k272147

edited Sep 5 at 23:53

Carl Woll

56.2k272147

56.2k272147

asked Sep 5 at 21:51

Isa

1276

asked Sep 5 at 21:51

Isa

1276

asked Sep 5 at 21:51

Isa

1276

1276

put on hold as off-topic by AccidentalFourierTransform, m_goldberg, Daniel Lichtblau, Henrik Schumacher, José Antonio Díaz Navas Sep 7 at 11:39

• The question does not concern the technical computing software Mathematica by Wolfram Research. Please see the help center to find out about the topics that can be asked here.

If this question can be reworded to fit the rules in the help center, please edit the question.

put on hold as off-topic by AccidentalFourierTransform, m_goldberg, Daniel Lichtblau, Henrik Schumacher, José Antonio Díaz Navas Sep 7 at 11:39

• The question does not concern the technical computing software Mathematica by Wolfram Research. Please see the help center to find out about the topics that can be asked here.

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Why should it? The right hand side is much more complicated than the left one.
  – Henrik Schumacher
  Sep 5 at 21:52
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Yes, it is the $Gamma$-function. But how on earth should Mathematica tell that you want to have the expression Gamma[2 z] expanded in exactly this way? For example, it could also return Gamma[2 z - 1] (2 z - 1). Or any other identity involving the $Gamma$-function (there are probably really many identities at least as interesting as the one you gave). What I tried to explain to you: i) Computers cannot read your mind and ii) Mathematica just doesn't work this way.
  – Henrik Schumacher
  Sep 5 at 22:06
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It's not a command. Don't think of it that way. It's an expression. Mathematica is an expression rewriting language. If you want it to rewrite an expression in a particular way, you must tell it.
  – John Doty
  Sep 5 at 22:59
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I'm voting to close this question as off-topic because the issued raised is not really a problem; it is arises from the OP's misunderstanding of the result returned by Mathematica.
  – m_goldberg
  Sep 5 at 23:16
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  That rhs looks like gamma after she got run over by a reindeer...
  – Daniel Lichtblau
  Sep 6 at 4:18
  

  
  
  
  

 | 
show 8 more comments

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Why should it? The right hand side is much more complicated than the left one.
  – Henrik Schumacher
  Sep 5 at 21:52
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Yes, it is the $Gamma$-function. But how on earth should Mathematica tell that you want to have the expression Gamma[2 z] expanded in exactly this way? For example, it could also return Gamma[2 z - 1] (2 z - 1). Or any other identity involving the $Gamma$-function (there are probably really many identities at least as interesting as the one you gave). What I tried to explain to you: i) Computers cannot read your mind and ii) Mathematica just doesn't work this way.
  – Henrik Schumacher
  Sep 5 at 22:06
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  It's not a command. Don't think of it that way. It's an expression. Mathematica is an expression rewriting language. If you want it to rewrite an expression in a particular way, you must tell it.
  – John Doty
  Sep 5 at 22:59
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I'm voting to close this question as off-topic because the issued raised is not really a problem; it is arises from the OP's misunderstanding of the result returned by Mathematica.
  – m_goldberg
  Sep 5 at 23:16
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  That rhs looks like gamma after she got run over by a reindeer...
  – Daniel Lichtblau
  Sep 6 at 4:18
  

  
  
  
  

3

3

Why should it? The right hand side is much more complicated than the left one.
– Henrik Schumacher
Sep 5 at 21:52

Why should it? The right hand side is much more complicated than the left one.
– Henrik Schumacher
Sep 5 at 21:52

3

3

Yes, it is the $Gamma$-function. But how on earth should Mathematica tell that you want to have the expression Gamma[2 z] expanded in exactly this way? For example, it could also return Gamma[2 z - 1] (2 z - 1). Or any other identity involving the $Gamma$-function (there are probably really many identities at least as interesting as the one you gave). What I tried to explain to you: i) Computers cannot read your mind and ii) Mathematica just doesn't work this way.
– Henrik Schumacher
Sep 5 at 22:06

Yes, it is the $Gamma$-function. But how on earth should Mathematica tell that you want to have the expression Gamma[2 z] expanded in exactly this way? For example, it could also return Gamma[2 z - 1] (2 z - 1). Or any other identity involving the $Gamma$-function (there are probably really many identities at least as interesting as the one you gave). What I tried to explain to you: i) Computers cannot read your mind and ii) Mathematica just doesn't work this way.
– Henrik Schumacher
Sep 5 at 22:06

4

4

It's not a command. Don't think of it that way. It's an expression. Mathematica is an expression rewriting language. If you want it to rewrite an expression in a particular way, you must tell it.
– John Doty
Sep 5 at 22:59

It's not a command. Don't think of it that way. It's an expression. Mathematica is an expression rewriting language. If you want it to rewrite an expression in a particular way, you must tell it.
– John Doty
Sep 5 at 22:59

2

2

I'm voting to close this question as off-topic because the issued raised is not really a problem; it is arises from the OP's misunderstanding of the result returned by Mathematica.
– m_goldberg
Sep 5 at 23:16

I'm voting to close this question as off-topic because the issued raised is not really a problem; it is arises from the OP's misunderstanding of the result returned by Mathematica.
– m_goldberg
Sep 5 at 23:16

5

5

That rhs looks like gamma after she got run over by a reindeer...
– Daniel Lichtblau
Sep 6 at 4:18

That rhs looks like gamma after she got run over by a reindeer...
– Daniel Lichtblau
Sep 6 at 4:18

 | 
show 8 more comments

2 Answers
2

active

oldest

votes

up vote
10
down vote



accepted

You can use an undocumented internal function to for this purpose:

Simplify`GammaTuplicate[Gamma[2z], 2] //TeXForm

$frac2^2 z-1 Gamma (z) Gamma left(z+frac12right)sqrtpi $

Another example:

Simplify`GammaTuplicate[Gamma[3z], 3] //TeXForm

$frac3^3 z-1/2 Gamma (z) Gamma left(z+frac13right) Gamma left(z+frac23right)2 pi $

share|improve this answer

answered Sep 5 at 23:50

Carl Woll

56.2k272147

add a comment | 

up vote
9
down vote

A top level way to look up the identity is through MathematicalFunctionData:

identity = MathematicalFunctionData[Gamma, "MultipliedArgumentFormulas", 
 "IncludedSubexpressions" -> Gamma[2 _]][[1]];

Activate[identity[z]]

Gamma[2 z] == (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

---

An obscure way of getting the identity is to take backward and forward Mellin transforms and to mix in some hackery that prevent simplifications along the way:

Block[Simplify`SimplifyGamma = # &,
 MellinTransform[MeijerGReduce[InverseMellinTransform[Gamma[2 z], z, s], s], s, z]
]

 (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

share|improve this answer

edited Sep 6 at 12:12

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
10
down vote



accepted

You can use an undocumented internal function to for this purpose:

Simplify`GammaTuplicate[Gamma[2z], 2] //TeXForm

$frac2^2 z-1 Gamma (z) Gamma left(z+frac12right)sqrtpi $

Another example:

Simplify`GammaTuplicate[Gamma[3z], 3] //TeXForm

$frac3^3 z-1/2 Gamma (z) Gamma left(z+frac13right) Gamma left(z+frac23right)2 pi $

share|improve this answer

answered Sep 5 at 23:50

Carl Woll

56.2k272147

add a comment | 

up vote
10
down vote



accepted

You can use an undocumented internal function to for this purpose:

Simplify`GammaTuplicate[Gamma[2z], 2] //TeXForm

$frac2^2 z-1 Gamma (z) Gamma left(z+frac12right)sqrtpi $

Another example:

Simplify`GammaTuplicate[Gamma[3z], 3] //TeXForm

$frac3^3 z-1/2 Gamma (z) Gamma left(z+frac13right) Gamma left(z+frac23right)2 pi $

share|improve this answer

answered Sep 5 at 23:50

Carl Woll

56.2k272147

add a comment | 

up vote
10
down vote



accepted

up vote
10
down vote



accepted

You can use an undocumented internal function to for this purpose:

Simplify`GammaTuplicate[Gamma[2z], 2] //TeXForm

$frac2^2 z-1 Gamma (z) Gamma left(z+frac12right)sqrtpi $

Another example:

Simplify`GammaTuplicate[Gamma[3z], 3] //TeXForm

$frac3^3 z-1/2 Gamma (z) Gamma left(z+frac13right) Gamma left(z+frac23right)2 pi $

share|improve this answer

answered Sep 5 at 23:50

Carl Woll

56.2k272147

You can use an undocumented internal function to for this purpose:

Simplify`GammaTuplicate[Gamma[2z], 2] //TeXForm

$frac2^2 z-1 Gamma (z) Gamma left(z+frac12right)sqrtpi $

Another example:

Simplify`GammaTuplicate[Gamma[3z], 3] //TeXForm

$frac3^3 z-1/2 Gamma (z) Gamma left(z+frac13right) Gamma left(z+frac23right)2 pi $

share|improve this answer

answered Sep 5 at 23:50

Carl Woll

56.2k272147

share|improve this answer

share|improve this answer

answered Sep 5 at 23:50

Carl Woll

56.2k272147

answered Sep 5 at 23:50

Carl Woll

56.2k272147

answered Sep 5 at 23:50

Carl Woll

56.2k272147

56.2k272147

add a comment | 

add a comment | 

up vote
9
down vote

A top level way to look up the identity is through MathematicalFunctionData:

identity = MathematicalFunctionData[Gamma, "MultipliedArgumentFormulas", 
 "IncludedSubexpressions" -> Gamma[2 _]][[1]];

Activate[identity[z]]

Gamma[2 z] == (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

---

An obscure way of getting the identity is to take backward and forward Mellin transforms and to mix in some hackery that prevent simplifications along the way:

Block[Simplify`SimplifyGamma = # &,
 MellinTransform[MeijerGReduce[InverseMellinTransform[Gamma[2 z], z, s], s], s, z]
]

 (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

share|improve this answer

edited Sep 6 at 12:12

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

add a comment | 

up vote
9
down vote

A top level way to look up the identity is through MathematicalFunctionData:

identity = MathematicalFunctionData[Gamma, "MultipliedArgumentFormulas", 
 "IncludedSubexpressions" -> Gamma[2 _]][[1]];

Activate[identity[z]]

Gamma[2 z] == (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

---

An obscure way of getting the identity is to take backward and forward Mellin transforms and to mix in some hackery that prevent simplifications along the way:

Block[Simplify`SimplifyGamma = # &,
 MellinTransform[MeijerGReduce[InverseMellinTransform[Gamma[2 z], z, s], s], s, z]
]

 (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

share|improve this answer

edited Sep 6 at 12:12

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

add a comment | 

up vote
9
down vote

up vote
9
down vote

A top level way to look up the identity is through MathematicalFunctionData:

identity = MathematicalFunctionData[Gamma, "MultipliedArgumentFormulas", 
 "IncludedSubexpressions" -> Gamma[2 _]][[1]];

Activate[identity[z]]

Gamma[2 z] == (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

---

An obscure way of getting the identity is to take backward and forward Mellin transforms and to mix in some hackery that prevent simplifications along the way:

Block[Simplify`SimplifyGamma = # &,
 MellinTransform[MeijerGReduce[InverseMellinTransform[Gamma[2 z], z, s], s], s, z]
]

 (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

share|improve this answer

edited Sep 6 at 12:12

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

A top level way to look up the identity is through MathematicalFunctionData:

identity = MathematicalFunctionData[Gamma, "MultipliedArgumentFormulas", 
 "IncludedSubexpressions" -> Gamma[2 _]][[1]];

Activate[identity[z]]

Gamma[2 z] == (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

---

An obscure way of getting the identity is to take backward and forward Mellin transforms and to mix in some hackery that prevent simplifications along the way:

Block[Simplify`SimplifyGamma = # &,
 MellinTransform[MeijerGReduce[InverseMellinTransform[Gamma[2 z], z, s], s], s, z]
]

 (2^(-1 + 2 z) Gamma[z] Gamma[1/2 + z])/Sqrt[[Pi]]

share|improve this answer

edited Sep 6 at 12:12

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

share|improve this answer

share|improve this answer

edited Sep 6 at 12:12

edited Sep 6 at 12:12

edited Sep 6 at 12:12

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

answered Sep 5 at 23:50

Chip Hurst

18.9k15484

18.9k15484

add a comment | 

add a comment |