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ratio_of_tgamma Class Reference

The class ratio_of_tgamma contains the ratio of two Gamma functions. More...

#include <polygamma.h>

List of all members.

Public Member Functions

 ratio_of_tgamma (const GiNaC::ex &a1, const GiNaC::ex &b1, const GiNaC::ex &a2, const GiNaC::ex &b2)
 ratio_of_tgamma (const GiNaC::ex &a1, const GiNaC::ex &b1, const GiNaC::ex &a2, const GiNaC::ex &b2, const GiNaC::ex &i, const GiNaC::ex &eps, int o, int f)
void archive (GiNaC::archive_node &node) const
void read_archive (const GiNaC::archive_node &node, GiNaC::lst &sym_lst)
unsigned return_type (void) const
void print (const GiNaC::print_context &c, unsigned level=0) const
unsigned precedence (void) const
GiNaC::ex eval (int level=0) const
GiNaC::ex subs (const GiNaC::exmap &m, unsigned options=0) const
virtual GiNaC::ex eval_explicit (int level=0) const
GiNaC::ex set_expansion (void) const
GiNaC::ex set_index (const GiNaC::ex &new_index) const
GiNaC::ex set_expansion_parameter (const GiNaC::ex &new_eps) const
GiNaC::ex set_order (int new_order) const
GiNaC::ex set_flag (int new_flag) const
GiNaC::ex set_values (const GiNaC::ex &new_index, const GiNaC::ex &new_eps, int new_order, int new_flag) const
GiNaC::ex get_index (void) const
GiNaC::ex shift_plus_one (void) const
GiNaC::ex shift_minus_one (void) const
GiNaC::ex index_eq_one (void) const
GiNaC::ex shift_index (const GiNaC::ex &new_index) const
int pole_alert (void) const
int expansion_alert (void) const

Protected Member Functions

GiNaC::ex eval_ncmul (const GiNaC::exvector &v) const
GiNaC::ex derivative (const GiNaC::symbol &s) const
unsigned calchash (void) const
GiNaC::ex increase_numerator (int shift_order=0) const
GiNaC::ex decrease_numerator (int shift_order=0) const
GiNaC::ex increase_denominator (int shift_order=0) const
GiNaC::ex decrease_denominator (int shift_order=0) const
GiNaC::ex prefactor_increase_numerator (void) const
GiNaC::ex prefactor_decrease_numerator (void) const
GiNaC::ex prefactor_increase_denominator (void) const
GiNaC::ex prefactor_decrease_denominator (void) const

Protected Attributes

GiNaC::ex a1
GiNaC::ex b1
GiNaC::ex a2
GiNaC::ex b2
GiNaC::ex index
GiNaC::ex expansion_parameter
int order
int flag_expand_status

Detailed Description

The class ratio_of_tgamma contains the ratio of two Gamma functions.

The definition is

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} \]

Here $ a_1 $, $ b_1 $, $ a_2 $ and $ b_2 $ are passed to the constructor. The index $i$ and the expansion parameter $\varepsilon$ can be set via the set_index(const ex &) and set_expansion_parameter(const ex &) methods, respectively.

Additionally, this class contains an internal variable "order", which indicates to which order this object should be expanded. The "order" variable can be set via the method set_order(int). The expansion is performed by calling the method set_expansion(void).

This class assumes that $i+a_1 > 0$ for all $i$, e.g. it does not handle possible poles in the Gamma function.

Member Function Documentation

ex decrease_denominator (  int  shift_order = 0  )  const [protected]
 

Returns

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2-1+b_2\varepsilon)} \]

The order variable is adjusted according to order = order + shift_order.

ex decrease_numerator (  int  shift_order = 0  )  const [protected]
 

Returns

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1-1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} \]

The order variable is adjusted according to order = order + shift_order.

ex eval (  int  level = 0  )  const
 

If the flag flag_expand_status is set, the object is expanded in $\varepsilon$.

If $a_1$ and $a_2$ are zero, we have

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+b_1\varepsilon)}{\Gamma(i+b_2\varepsilon)} = \frac{ 1 + b_1 \varepsilon Z_1(i-1) + ...}{1 + b_2 \varepsilon Z_1(i-1) + ...} \]

Here the formula

\[ \left( 1 + \varepsilon Z_1(i-1) + \varepsilon^2 Z_{11}(i-1) + ... \right)^{-1} = 1 - \varepsilon S_1(i-1) + \varepsilon^2 S_{11}(i-1) - ... \]

is useful.

If $a_1 > 0$ we have

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} = \left( i + a_1 -1 + b_1 \varepsilon \right) \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1-1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} \]

$i$ is interpreted as unit_letter(-1,index). The order parameter of ratio_of_tgamma of the $b_1 \varepsilon$ term is decreased by one due to the explicit presence of $\varepsilon$.

If $a_1 < 0$ we have

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} = \sum\limits_{j=0}^\infty \frac{1}{(i+a_1)^{j+1}} \left(-b_1\right)^j \varepsilon^j \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} \]

The order parameter is adjusted according to the prefactors of $\varepsilon$. Note that this formula might give rise to poles for $i+a_1=0$. These possible poles have to be dealt with at a higher level.

If $a_2 > 0$ we have

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} = \sum\limits_{j=0}^\infty \frac{1}{(i+a_2-1)^{j+1}} \left(-b_2\right)^j \varepsilon^j \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2-1+b_2\varepsilon)} \]

The order parameter is adjusted according to the prefactors of $\varepsilon$.

If $a_2 < 0$ we have

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} = \left( i + a_2 + b_2 \varepsilon \right) \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+1+b_2\varepsilon)} \]

The order parameter is adjusted according to the prefactors of $\varepsilon$.

The return type is either Zsum or basic_letter * Zsum. If $a_1 \ge 0 $ we do not get any poles.

If the index is an integer and the status flag not equal to no_eval_to_scalar, the object is evaluated explicitly.

ex eval_explicit (  int  level = 0  )  const [virtual]
 

Explicit evaluation with the help of nestedsums_helper_expand_tgamma_fct.

ex eval_ncmul (  const GiNaC::exvector &  v  )  const [protected]
 

No automatic simplifications

int expansion_alert (  void   )  const
 

Returns 1 if $a_1$ or $a_2$ are not integers, and 0 otherwise.

ex get_index (  void   )  const
 

Returns the index.

ex increase_denominator (  int  shift_order = 0  )  const [protected]
 

Returns

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+b_1\varepsilon)}{\Gamma(i+a_2+1+b_2\varepsilon)} \]

The order variable is adjusted according to order = order + shift_order.

ex increase_numerator (  int  shift_order = 0  )  const [protected]
 

Returns

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+1+b_1\varepsilon)}{\Gamma(i+a_2+b_2\varepsilon)} \]

The order variable is adjusted according to order = order + shift_order.

ex index_eq_one (  void   )  const
 

Returns the expansions of

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(a_1+1+b_1 \varepsilon)}{\Gamma(a_2+1+b_2 \varepsilon)} \]

up to order + shift_order.

int pole_alert (  void   )  const
 

Returns 1 if $a_1 < 0 $, and 0 otherwise.

ex prefactor_decrease_denominator (  void   )  const [protected]
 

Returns

\[ \sum\limits_{j=0}^\infty \frac{1}{(i+a_2-1)^{j+1}} \left(-b_2\right)^j \varepsilon^j \]

ex prefactor_decrease_numerator (  void   )  const [protected]
 

Returns

\[ \left(i + a_1-1 \right) +b_1 \varepsilon i^0 \]

ex prefactor_increase_denominator (  void   )  const [protected]
 

Returns

\[ i + a_2 i^0 + b_2 \varepsilon i^0 \]

ex prefactor_increase_numerator (  void   )  const [protected]
 

Returns

\[ \sum\limits_{j=0}^\infty \frac{1}{(i+a_1)^{j+1}} \left(-b_1\right)^j \varepsilon^j \]

ex set_expansion (  void   )  const
 

Sets the flag expansion_required. The object is then automatically expanded up to the order specified in the member variable order.

ex set_expansion_parameter (  const GiNaC::ex &  new_eps  )  const
 

Sets the member variable expansion_parameter to new_eps.

ex set_flag (  int  new_flag  )  const
 

Sets the member variable flag_expand_status to new_flag

ex set_index (  const GiNaC::ex &  new_index  )  const
 

Sets the member variable index to new_index.

ex set_order (  int  new_order  )  const
 

Sets the member variable order to new_order.

ex set_values (  const GiNaC::ex &  new_index,
const GiNaC::ex &  new_eps,
int  new_order,
int  new_flag
)  const
 

Sets the member variables index, expansion_parameter and order to new values.

ex shift_index (  const GiNaC::ex &  new_index  )  const
 

Returns a ratio_of_tgamma with

\[ \Gamma(i + a + o + b \varepsilon ) \]

with $ o = j-i$, where $i$ is the new index and $j$ is the old index.

ex shift_minus_one (  void   )  const
 

Returns

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1-1+b_1 \varepsilon)}{\Gamma(i+a_2-1+b_2 \varepsilon)} \]

The order parameter is not changed.

ex shift_plus_one (  void   )  const
 

Returns

\[ \frac{\Gamma(1+b_2\varepsilon)}{\Gamma(1+b_1\varepsilon)} \frac{\Gamma(i+a_1+1+b_1 \varepsilon)}{\Gamma(i+a_2+1+b_2 \varepsilon)} \]

The order parameter is not changed.

The documentation for this class was generated from the following files:
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