a
    kº”hj  ã                   @   sN  d dl mZmZmZmZmZmZmZmZm	Z	m
Z
mZmZmZmZmZmZmZ d dlmZmZ d dlmZmZ d dlmZ d dlmZmZmZmZ d dlm Z  d dl!m"Z"m#Z# d dl$m%Z% d d	l&m'Z'm(Z( d d
l)m*Z*m+Z+m,Z, d dl-m.Z. d dl/m0Z0m1Z1m2Z2m3Z3 d dl4m5Z5m6Z6m7Z7m8Z8 d dl9m:Z: d dl;m<Z<m=Z= d dl>m?Z? d dl@mAZA d dlBmCZC d dlDmEZE d dlFmGZG d dlHmIZImJZJmKZKmLZL d dlMmNZNmOZOmPZPmQZQmRZRmSZS e#dƒ\ZTZUZVdd„ ZWdd„ ZXdd„ ZYeJeIdd„ ƒƒZZd d!„ Z[d"d#„ Z\eJd$d%„ ƒZ]eJd&d'„ ƒZ^eJd(d)„ ƒZ_d*d+„ Z`d,d-„ Zad.d/„ Zbd0d1„ Zcd2d3„ Zdd4d5„ Zed6d7„ Zfd8S )9é    )Úmellin_transformÚinverse_mellin_transformÚfourier_transformÚinverse_fourier_transformÚsine_transformÚinverse_sine_transformÚcosine_transformÚinverse_cosine_transformÚhankel_transformÚinverse_hankel_transformÚFourierTransformÚSineTransformÚCosineTransformÚInverseFourierTransformÚInverseSineTransformÚInverseCosineTransformÚIntegralTransformError)Úlaplace_transformÚinverse_laplace_transform)ÚFunctionÚ
expand_mul)Ú
EulerGamma)ÚIÚRationalÚooÚpi)ÚS)ÚSymbolÚsymbols)Ú	factorial)ÚreÚ
unpolarify)ÚexpÚ	exp_polarÚlog)Úsqrt)ÚatanÚcosÚsinÚtan)ÚbesseliÚbesseljÚbesselkÚbessely)Ú	Heaviside)ÚerfÚexpint)Úgamma)Úmeijerg)Ú	gammasimp)Úhyperexpand)Útrigsimp)ÚXFAILÚslowÚskipÚraises)ÚxÚsÚaÚbÚcÚdznu beta rhoc                  C   s„   ddl m}  tdƒ}t|tƒttƒ| |tƒttƒks8J ‚t|tƒtt ƒ ttƒ| |tƒttƒttd ƒt  dtfdfks€J ‚d S )Nr   )ÚMellinTransformÚfé   T)	Úsympy.integrals.transformsr@   r   r   r:   r;   r"   r1   r   )r@   rA   © rD   úS/var/www/auris/lib/python3.9/site-packages/sympy/integrals/tests/test_transforms.pyÚtest_undefined_function$   s    $(ÿrF   c                  C   sJ   t dƒ} t| tƒttƒjthks$J ‚t| tƒt ttƒjtthksFJ ‚d S )NrA   )r   r   r:   r;   Zfree_symbolsr<   )rA   rD   rD   rE   Útest_free_symbols,   s    rG   c                  C   sŠ  ddl m}  tdƒ}t|tƒttƒ d¡| ttd  |tƒ tdtfƒksNJ ‚t|tƒttƒ d¡| |tƒt	dt
 t t t ƒ tt tfƒks–J ‚t|tƒttdd d¡| |tƒt	t t ƒ tdtfƒksÖJ ‚td	t t
 t|tƒttttfƒ d¡ ƒd
ksJ ‚td	t t
 t|tƒttƒ d¡ ƒdks<J ‚t|tƒttƒ d¡| |tƒt	d	t
 t t t ƒ tt tfƒks†J ‚d S )Nr   )ÚIntegralrA   rH   rB   éþÿÿÿT©Znocondsé   z.Integral(f(s)/x**s, (s, _c - oo*I, _c + oo*I))z2Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I)))Zsympy.integrals.integralsrH   r   r   r:   r;   Úrewriter   r   r"   r   r   r   Ústrr   r<   r=   r   r   )rH   rA   rD   rD   rE   Útest_as_integral2   s(    ÿ,ÿ ÿ*ÿ
$ÿ
,ÿrN   c                  C   sà  t dƒ t} tddd}tttd  ƒt t tttd  ƒ }| | t| ¡ttƒdtd  dtdt    |tdt  d   t	tt ƒ t	dt dt  ƒ t	dt ƒ t
tƒ t
tƒ d tj fdfksÖJ ‚tttd  ƒt t }| | t| ¡ttƒdtdt   t |tdt    t	t dt  ƒ t	tt ƒ t	t d ƒ t
tƒ t
tƒ d fdfkstJ ‚| | t| tdi¡ttƒ|dt d   t	tƒ t	t tj ƒ dttƒ  dtddƒfdfksÜJ ‚d S )NzRisch takes forever.r=   T©ZpositiverK   éÿÿÿÿrB   )r8   r   r   r%   r:   r=   r<   Úsubsr;   r1   r    r   ÚHalfr   r   )ÚMTÚbposÚexprrD   rD   rE   Útest_mellin_transform_failE   s<    (:ÿ
ÿþÿ*ÿ
ÿÿýÿ
4ÿÿrV   c                  C   s¼  ddl m} m} t}tddd}|tt ttd ƒ ttƒdtt  t	 t
tƒ fdfks^J ‚|tt tdt ƒ ttƒdtt  t
tƒ t	fdfksšJ ‚|dt td  tdt ƒ ttƒttƒttƒ ttt ƒ dt	ft
tƒdkfksðJ ‚|td td  ttd ƒ ttƒttƒtdt t ƒ tdt ƒ t	 dt
tƒ ft
tƒdkfksZJ ‚|dt t  ttƒttƒttt ƒ ttƒ dt
tƒfdfks J ‚|tdt ƒt  ttƒdttt d ƒ tdt ƒ ttttd   ƒ ttƒ ttt ƒ t dt
tƒft
tƒdk fks"J ‚|dt td  tdt ƒ ttd td   ttd ƒ  ttƒ}|d sšJ |d dt
tƒ d ft
tƒdkfkƒ‚|tt tt  tt  ttƒd tttt d   ttt ƒ ttt ƒtttt  ƒ  ks J ‚|tt |t  t|  ttƒt|tt d   ttt ƒ ttt ƒtttt  ƒ  | dt
tƒ ƒ|ddt
tƒ ƒfdfks†J ‚tttd  ƒt t }|| t|¡ttƒt d| tdt    ttƒ tt dt  ƒ tt t d ƒ dt
tƒ d fdfksJ ‚tttd  ƒt t tttd  ƒ }|| t|¡ttƒdtdt   |tdt  d   ttƒ tdt dt  ƒ tdt t ƒ dt
tƒ d tj fdfks¾J ‚|tt ƒttƒttƒdt	fdfksèJ ‚|tdt ƒttƒtt ƒt	 dfdfksJ ‚|ttƒd	 tdt ƒ ttƒd
td  dt	fdfksTJ ‚|ttƒd ttd ƒ ttƒdtd	  t	 dfdfks’J ‚|ttd ƒttƒttttt ƒ  ddfksÆJ ‚|tdt d ƒttƒttttt ƒ  ddfksþJ ‚|ttdt ƒƒttƒttttt ƒ  ddfks6J ‚|ttddt  ƒƒttƒttttt ƒ  ddfksrJ ‚|tttƒƒttƒtttj ƒ ttƒt  tddƒdfdfks¸J ‚d S )Nr   ©ÚMaxÚMinr=   TrO   rB   rP   rK   é   é   é   é   é   ©rP   r   ©r   rB   )Ú(sympy.functions.elementary.miscellaneousrX   rY   r   r   r:   Únur.   r;   r   r    Úbetar1   ÚrhoÚabsr(   r   r'   r<   r=   r%   rQ   r   rR   r"   r$   r)   r/   r   )rX   rY   rS   rT   ÚmtrU   rD   rD   rE   Útest_mellin_transformc   s’    ÿÿ",ÿ""ÿÿ
(ÿ
ÿÿ
ÿÿýÿÿ4":ÿ
:"ÿÿ
Dÿÿ
(*ÿÿþÿ
*0<>488<*ÿrg   c                  C   sâ   t } | ttƒtd  ttƒ}|dd … dks0J ‚t|d dd t¡rJJ ‚| ttƒd td  ttƒ}|dd … dkszJ ‚t|d dd t¡r”J ‚| ttƒtd d  ttƒ}|dd … dksÄJ ‚t|d dd t¡rÞJ ‚d S )NrB   )r`   Tr   T)Zallow_hyperrK   ))r   rK   T)r   r$   r:   r;   r4   Zhasr2   )rS   rf   rD   rD   rE   Útest_mellin_transform2¢   s    rh   c               	   C   sœ  ddl m}  t}|ttdttƒ ƒttƒttd t ƒttd t d ƒ t	tƒ d t
ddƒfdfkslJ ‚|tttƒƒttttƒƒ ttƒdt tdt tj ƒ ttd t tj ƒ tt d t d ƒttdt  d ƒ  t	tƒ d tj t
ddƒfdfksJ ‚|tttƒƒttttƒƒ ttƒdt ttd t ƒ tdt tj ƒ tt d t tj ƒttdt  d ƒ  t	tƒ d t
ddƒfdfks¦J ‚|ttttƒƒd ttƒttt ƒttjt ƒ ttƒtdt ƒ tdt t ƒ  t	tƒ tjfdfksJ ‚|ttttƒƒtt ttƒƒ ttƒttƒttjt ƒ ttƒtdt t ƒ tdt t ƒ  dtjfdfksŒJ ‚|ttd ttƒƒttttƒƒ ttƒtdt ƒttt tj ƒ ttƒtt
ddƒt ƒ ttt tj ƒ  tjt	tƒ tjfdfksJ ‚|ttttƒƒttttƒƒ ttƒdt tddt  ƒ ttt d t ƒ tdt tt d  ƒtdt tt d  ƒ tdt tt d  ƒ  t	tƒt	tƒ  d tjfdfksÖJ ‚|ttttƒƒd tt ttƒƒd  ttƒdd … | t	tƒt	tƒ ƒtjfdfks0J ‚|ttdttƒ ƒttƒtttd t  ƒ tttd  ƒ tttd  ƒ t | t	tƒ d t	tƒd ƒt
ddƒfdfks²J ‚|tttƒƒttttƒƒ ttƒdt  tttd t  ƒ ttjdt  ƒ tdt d t ƒ tdt d t ƒ ttƒtdt td  ƒ tdt td  ƒ  | t	tƒd  d t	tƒd d ƒt
ddƒfdfks’J ‚|tttƒƒttttƒƒ ttƒdt  tttd t  ƒ tttd  ƒ tttd  ƒ ttjdt  ƒ ttƒttjt td  ƒ ttjt td  ƒ  | t	tƒ d t	tƒd ƒt
ddƒfdfksfJ ‚|ttttƒƒttttƒƒ ttƒttt ƒ ttƒ ttt ƒ ttjt ƒ ttd	ƒ tdt t ƒ  | t	tƒ dƒtjfdfksôJ ‚|ttttƒƒttttƒƒ ttƒdt  tttd td  t  ƒ tddt  ƒ ttd td  t ƒ ttd td  t ƒ tttd td  t d ƒ ttd td  t d ƒ  | t	tƒ t	tƒ d t	tƒ t	tƒ d ƒtjfdfksöJ ‚|ttttƒƒd ttƒdd … | t	tƒ dt	tƒƒtjfdfks>J ‚|ttdttƒ ƒttƒtttd  ƒtttd  ƒ d | t	tƒ d t	tƒd ƒtfdfks¤J ‚|ttdtdttƒ ƒ ƒttdtdttƒ ƒ ƒ ttƒdt  tdt ƒ ttd t ƒ dttd t d ƒ  | dt	tƒ d ƒtfdfk	s>J ‚|ttttƒƒttttƒƒ ttƒttƒttt ƒ tt tj ƒ dttƒ ttt d ƒ  | t	tƒ dƒtjfdfk	sÀJ ‚|ttttƒƒttttƒƒ ttƒddt d  tdt d ƒ tt d td  t ƒ ttd td  t ƒ tt d td  t d ƒttd td  t d ƒ  | t	tƒ d t	tƒd  t	tƒd t	tƒd  ƒtjfdfk
s²J ‚|tt d ƒtttd ƒ ttƒ}ttt|d jdd
ƒƒƒ}|dtt
ddƒ  ttt ƒ ttjt ƒ tdt t ƒtdt t ƒ tt t d ƒ ttt d ƒ  ksjJ ‚|dd … | t	tƒ t	tƒƒtfdfks˜J ‚d S )Nr   ©rX   rK   rB   r]   rZ   TrI   z3/2©Úfunc)ra   rX   r   r+   r<   r%   r:   r;   r1   r    r   r(   r   rR   r'   r   r=   r-   r,   r   r*   r"   r3   r5   Úexpand)rX   rS   rf   Zmt0rD   rD   rE   Útest_mellin_transform_bessel²   s:   <ÿ .(ÿÿþÿ
 (*ÿÿþÿ
"ÿþÿ
$&ÿ
þÿ
&*ÿþÿ
"*.ÿÿýÿ
4ÿ
8&ÿÿ
 .ÿÿ.þ.ýÿ
 N2ÿ&þÿ
".ÿþÿ
"4ÿÿ:þ4ýÿ
 ÿ

ÿÿÿ ÿÿ
ÿÿÿÿÿÿ
"ÿÿÿþÿ
"8ÿÿÿÿÿýÿ
$,@ÿrm   c                  C   s†  ddl m}  ddlm}m}m} ddlm} tddd}tddd	}t	|t
ƒt
tƒttƒt dtfdfksnJ ‚tttƒt tt
dtfƒ t¡ ¡ |t
ƒksžJ ‚t	ttt
ƒt
tƒttƒtt d
  | d
ttƒ dƒtfdfksàJ ‚|ttttƒ|t d
  tt
d
| tfƒ t¡jddƒƒt|t
ƒks,J ‚t	|t
ƒt
tƒdt  ttƒ ttd tj ƒ dt tt d d
 ƒ  ddfks†J ‚tdt  ttƒ ttd
 d ƒ dt tt d d
 ƒ  tt
dƒ|t
ƒksÚJ ‚t	|tt
ƒƒt
tƒddt d
   ttƒ ttƒ ttt tj ƒ  ddfks0J ‚tdt  ttƒ ttƒ dt tt tj ƒ  t|dƒ ¡ |t|ƒƒks‚J ‚d S )Nr   ri   )ÚCiÚE1ÚSi©Úsimplifyr<   T©ÚnegativeÚu)ZpolarrB   rj   rK   r_   r`   rZ   )ra   rX   Ú'sympy.functions.special.error_functionsrn   ro   rp   Úsympy.simplify.simplifyrr   r   r   r:   r;   r1   r   r   rL   r0   rl   r<   r    r!   r%   r   r   rR   )rX   rn   ro   rp   rr   Zanegru   rD   rD   rE   Útest_expint  sd    *ÿÿÿ*ÿ
ÿÿÿÿý
"ÿÿÿ
"ÿÿþ
:ÿ
0þ
þrx   c               
      sˆ  ddl m}  ddlm}m} ddlm} ddlm‰  ddl	m
‰ t}|ttƒttdtfƒtt ƒkshJ ‚|tt ƒttt dfƒtdt ƒks’J ‚ˆ|tdtd  d  ttdtfƒƒtd d	 td	t ƒ d
t  ksÜJ ‚|d	td d	  ttdƒt tt d	 ƒ d ttd	 ƒdt   ks(J ‚|d	td d	  ttdƒt tt d	 ƒ d ttd	 ƒdt   kstJ ‚|ttƒttd	 ƒ ttd	tfƒtd	 tt ƒ t ks²J ‚tddd}|d	td d	  ttt ƒd tfƒ t|¡ t¡ 
¡ t|ƒtd	t| ƒ ƒ ksJ ‚tddd\}}||t |  tt| ƒ t ttdtfƒt| t|  ƒkshJ ‚|t|| t|  ƒ|t  tt| tfƒt| tt|  ƒ ks°J ‚‡ ‡fdd„}tddd}	|d|	t  ttt d fƒt|	 ttd	 ƒ ks J ‚|d	|	t  ttd tfƒt|	 td	t ƒ ks4J ‚||ttƒttƒ ttt ƒ ttdtfƒƒd	t td	  td	t ƒ ks„J ‚||ttƒtd	t t ƒ td	t ƒ ttt d fƒƒtd	 td	  ttd	 ƒ ksÞJ ‚||ttƒttt ƒ ttƒ ttdƒƒd	td	  t ksJ ‚|tt ttd	   ttt ƒ ttƒ ttt ƒ td	t ƒ td	t t ƒ t tt|ttƒ dƒ|d	ttƒ d	ƒfƒ}
ˆ t|
ddƒ tt¡ 
¡ tt  tt  t t  ksØJ ‚ˆ|d	t tƒ t d  ttƒ td	t d t ƒ tt d t ƒ td	t t ƒ ttdttƒ d fƒƒd	t td	 ƒ t ksbJ ‚ˆ|dt!dt   t"t!dt  d	   ttƒ td	t! dt  ƒ td	t! t ƒ ttdtt!ƒ d	 d fƒƒt"t!d	  t"d t d	tt"d   ƒd	 t!  tt d	tt"d   ƒd	 t!d	     t"d t  ks8J ‚ˆ|dtdt    t t"tdt    ttƒ tt dt  ƒ tt t d	 ƒ ttdttƒ d fƒƒt"t t d	tt"d   ƒd	 t  ksÐJ ‚|dtd  ttdtfƒt#tƒd
 td	t ƒ ksJ ‚| |dtd
  ttt dfƒddt#tƒd ttd	 ƒ ksJJ ‚|ttttt ƒ  ttdƒt#td	 ƒkszJ ‚|ttttt d ƒ  ttdƒt#td d	 ƒks²J ‚|tttdt t ƒ  ttt$ddƒdfƒt#t tƒd	 ƒksôJ ‚|ttttt ƒ  ttdƒt#d	d	t  ƒks(J ‚d d!„ }|||ttt%tt ƒ  ttdƒƒƒt#d	t ƒtd	t ƒ t#td	 ƒttd	 ƒ  t#tƒttd	 ƒ t#d	d	t  ƒttd	 ƒ  t#t d	 ƒtt d	 ƒ  fv sØJ ‚||t|tt ƒ t ttdƒƒt#d	t d	 ƒtd	t ƒ t#d	d	t  ƒttd	 ƒ  t#tƒ tt d	 ƒ t#d	d	t  ƒttd	 ƒ  t#t d	 ƒtt d	 ƒ  fv 	sˆJ ‚|ttt&j' ƒ t tƒt  ttt$ddƒdfƒt(t tƒƒk	sÊJ ‚ˆ|tt!d t ƒtt!d t d	 ƒ tttt!ƒ d t$dd
ƒfƒƒt)t!dt tƒ ƒk
s(J ‚ˆ|dt! tt&j'dt  ƒ ttt!d	 d  ƒ td	t t!d  ƒtd	dt  t! ƒ  tttt!ƒd	  d t$d	d
ƒfƒƒtt tƒƒt)t!t tƒƒ k
sÄJ ‚ˆ|dt! tt!d t ƒ tt&j'dt  ƒ tt&j't t!d  ƒtd	dt  t! ƒ  tttt!ƒ d t$d	d
ƒfƒƒt*t tƒƒt)t!t tƒƒ ksZJ ‚ˆ|tt!t ƒtt&j't ƒ t tƒtd	t ƒ td	t! t ƒ  tttt!ƒ t&j'fƒƒt)t!t tƒƒd ksÊJ ‚ˆ|ttƒtt&j't ƒ t tƒtd	t t! ƒ td	t! t ƒ  ttdt&j'fƒƒt)t! t tƒƒt)t!t tƒƒ ks@J ‚ˆ|d
t td"t d	 ƒ tt!d t"d  t ƒ tt! d t"d  t d	 ƒtt!d t"d  t d	 ƒ tt!d t"d  t d	 ƒ  tttt!ƒtt"ƒ  d t&j'fƒƒt)t!t tƒƒt)t"t tƒƒ ksJ ‚ˆ|ddt   t*tt! d tt" d  tt  ƒ td"t d	 ƒ tt!d t"d  t ƒ tt!d t"d  t ƒ ttt!d t"d  t d	 ƒ tt!d t"d  t d	 ƒ  tt|tt!ƒ d tt"ƒd  tt!ƒ d tt"ƒd  ƒt&j'fƒƒt)t!t tƒƒt)t" t tƒƒt)t"t tƒƒt*tt" ƒ    ttt" ƒ ksNJ ‚|tt*tt ƒ ttdt&j'fƒt tƒtd	  ks„J ‚d S )#Nr   ©rl   rW   )Úcot©Úpowsimprq   rP   rK   rB   rZ   )rP   N©NrB   ÚrT©Úrealza brO   c                    s    ˆˆ t | ddddƒ tt¡S )NF©ÚdeepT©Úforce)r   Úreplacer#   r"   )rU   ©r|   rr   rD   rE   Ú	simp_powsW  s    z0test_inverse_mellin_transform.<locals>.simp_powsrb   )r   NFr   r[   r\   r^   rƒ   r]   r_   )rI   r   r`   c                 S   sJ   ddl m} ddlm} ddlm} |||| ddddddd tt¡S )Nr   ry   r{   )Ú
logcombineTrƒ   )r„   r‚   )	Úsympy.core.functionrl   Úsympy.simplify.powsimpr|   rw   rˆ   r…   r#   r"   )rU   rl   r|   rˆ   rD   rD   rE   Úmysimpƒ  s    þþz-test_inverse_mellin_transform.<locals>.mysimprI   )+r‰   rl   ra   rX   rY   Ú(sympy.functions.elementary.trigonometricrz   rŠ   r|   rw   rr   r   r1   r;   r:   r   r"   r.   r   rQ   rL   r(   r   rc   rd   r?   r>   r   r    r   r…   r#   r%   r<   r=   r$   r   r)   r   rR   r/   r+   r'   )rl   rX   rY   rz   ZIMTr~   Z_aÚ_br‡   rb   rU   r‹   rD   r†   rE   Útest_inverse_mellin_transform5  sX   $*$ÿ*ÿ
*ÿ
>$ÿÿþ
DH64,ÿ
&ÿþ
(ÿ
 ÿ
ÿ
ÿÿÿ$þÿ
4ÿÿþý
BÿÿDÿ
ÿþ
Bÿÿ"þ
8"ÿ
08B4$.4ÿÿþ 68ÿÿþ.
ÿ
Dÿ
0&ÿþý
,(ÿþý
"ÿþý
&ÿþý
28ÿÿýü
Dÿÿ:þ:üÿ
ÿû
rŽ   c            
         s¬  ddl m‰ m‰m‰ ddlm}  ddlm‰ t}t	}‡ ‡‡‡fdd„}dd„ }t
d	d
d}tdƒ}t
dd
d}t
dd
d}t
dd
d}	t|tƒt|ƒt|tƒt|ƒksªJ ‚t	||ƒ|tƒt||ƒ|tƒksÎJ ‚||tdtd| t ƒ ƒt|ƒƒ||| ƒ| ksJ ‚||tdt|t ƒ ƒdt|t ƒ  t|ƒƒ||| ƒd | ksRJ ‚| |t| t ƒttƒ t|ƒtdd|dt t |   ks–J ‚|d|dt t t   t|	ddt| |	 ƒd
fksÒJ ‚|d|dt t t   t|	 dddksJ ‚|d|dt t t   tt
d	d
ddddks8J ‚| |tt| t ƒ ttƒ t|ƒtdd|dt t |  d  ks„J ‚|t| t ƒt|t ƒ ttƒ t|ƒ||d |dt t |  d   ksØJ ‚|t| td  ƒt|ƒttƒttd  |d  | ƒ t|ƒ ks"J ‚|tt| ƒtt| d  | ƒ |tƒt| td  ƒksdJ ‚|t| ttƒ ƒt|ƒd| |d dtd  |d    ks¨J ‚d S )Nr   )rl   Úexpand_complexÚexpand_trig)Úfactorrq   c                    s   ˆˆˆˆ | ƒƒƒƒS ©NrD   ©r:   ©rl   r   r   rr   rD   rE   ÚsimpÊ  s    z$test_fourier_transform.<locals>.simpc                 S   s   t t|  ƒt|   S r’   )r(   r   r“   rD   rD   rE   ÚsincÍ  s    z$test_fourier_transform.<locals>.sincÚkTr   rA   r<   rO   r=   ÚposkrB   rK   )Ú	extensionFrJ   )r   Trs   rZ   )r‰   rl   r   r   Zsympy.polys.polytoolsr‘   rw   rr   r   r   r   r   r:   r   r   r.   re   r"   r   r   r(   r%   )
r‘   ZFTZIFTr•   r–   r—   rA   r<   r=   r˜   rD   r”   rE   Útest_fourier_transformÃ  s^    $
ÿÿ:J$ÿ
ÿÿ
ÿÿ
$ÿÿ
(ÿ
("ÿ
JBrš   c                  C   s  t dƒ} t dƒ}t dƒ}tdƒ}t|| ƒ| |ƒt|| ƒ| |ƒksDJ ‚t||ƒ|| ƒt||ƒ|| ƒkshJ ‚tdt| ƒ | |ƒdt|ƒ ksŒJ ‚tdt|ƒ || ƒdt| ƒ ks°J ‚tdt| ƒ d | |ƒdt|ƒ ksØJ ‚t| |  | |ƒd| tj  ||d   t	| d d ƒ t	|d d ƒ ks.J ‚td| tdƒd   ||d   t	| d d ƒ t	|d tj ƒ || ƒ| |  ksŒJ ‚tt
| |  ƒ| |ƒtdƒ| ttƒ|d |d    ksÌJ ‚ttdƒ| ttƒ|d |d    || ƒt
| |  ƒksJ ‚tt| ƒ|  | |ƒtdƒttƒ t|d ƒdt    d ksPJ ‚t| t
| | d  ƒ | |ƒtdƒ| t
|d  d|  ƒ d|tddƒ   ks¨J ‚ttdƒ| t
|d  d|  ƒ d|tddƒ   || ƒ| t
| | d  ƒ ks J ‚d S )	NÚtÚwr<   rA   rB   r]   rK   rZ   )r   r   r   r   r   r   r%   r   rR   r1   r"   r   r$   r   r   )r›   rœ   r<   rA   rD   rD   rE   Útest_sine_transformý  s|    $
ÿÿ$$(
ÿ
ÿÿÿ
ÿÿ
ÿÿÿÿÿ
ÿ"ÿ
&ÿÿ
ÿ(ÿ
ÿ2ÿ
6ÿÿr   c                  C   s.  ddl m} m} tdƒ}tdƒ}tdƒ}tdƒ}t||ƒ||ƒt||ƒ||ƒksTJ ‚t||ƒ||ƒt||ƒ||ƒksxJ ‚tdt	|ƒ ||ƒdt	|ƒ ksœJ ‚tdt	|ƒ ||ƒdt	|ƒ ksÀJ ‚td|d |d   ||ƒt	dƒt	t
ƒ t| | ƒ d|  ksJ ‚t||  ||ƒd| tj  ||d   t| d d ƒ t|d ƒ ksZJ ‚td| tdƒd   ||d   t| d tj ƒ t|d ƒ ||ƒ||  ks´J ‚tt| | ƒ||ƒt	dƒ| t	t
ƒ|d |d    ksôJ ‚tt	dƒ| t	t
ƒ|d |d    ||ƒt| | ƒks4J ‚tt| t	|ƒ ƒt|t	|ƒ ƒ ||ƒ|t|d  d|  ƒ d|td	dƒ   ksJ ‚td||  ||ƒt	dƒd
||| ƒ t
 t|| ƒ d t|| ƒ| || ƒ   t	t
ƒ ksöJ ‚tt	dƒttjdfdftjddftjff|d |d  d ƒ dt
  ||ƒd||  ksVJ ‚tdt	|d |d  ƒ ||ƒt	dƒttjfdfdtjff|d |d  d ƒ dt	t
ƒ  ks¼J ‚tt	dƒttjfdfdtjff|d |d  d ƒ dt	t
ƒ  ||ƒd|t	|d |d  d ƒ  ks*J ‚d S )Nr   )rn   rp   r›   rœ   r<   rA   rB   rK   r]   rI   rD   rZ   )r   r   )rv   rn   rp   r   r   r   r   r	   r   r%   r   r"   r   rR   r1   r'   r   r(   r2   )rn   rp   r›   rœ   r<   rA   rD   rD   rE   Útest_cosine_transform   s¢    $
ÿÿ$$ÿÿ$ÿ
ÿÿ8ÿ

ÿÿ
ÿÿ
ÿÿÿ
ÿ"ÿ
&ÿÿ
ÿÿ*ÿ
:ÿÿÿÿÿÿ
ÿ
&&ÿ
ÿrž   c                  C   sØ  t dƒ} t dƒ}t dƒ}t dƒ}tdƒ}td|  | |dƒd| ksFJ ‚td| || dƒd|  ksdJ ‚td| |  | |dƒd| d  ||d   t| d d ƒ t|d ƒ ks¶J ‚td| d  ||d   t| d d ƒ t|d ƒ || dƒ| |  ksJ ‚td| |  | ||ƒdd|   ||d   t| d |d  d ƒ t|d |d  ƒ kslJ ‚td| d  ||d   t| d |d  d ƒ t|d |d  ƒ || |ƒ| |  ksÎJ ‚t| | t| |  ƒ | ||ƒd|d  | || d	   |d |d  d | td	ƒd    t|td	dƒ ƒ tt	ƒ ksRJ ‚td|d  | || d	   |d |d  d | td	dƒ   t|td	dƒ ƒ tt	ƒ || |ƒ| | t| |  ƒ ksÔJ ‚d S )
Nr~   r—   rb   Úmr<   rB   r   rK   r]   )
r   r   r
   r   r1   r"   r   r   r%   r   )r~   r—   rb   rŸ   r<   rD   rD   rE   Útest_hankel_transformJ  sh    ÿ6ÿ<ÿÿ
Fÿ
ÿÿÿÿÿ
4ÿÿÿÿÿ
Bÿÿþþr    c                   C   s    t ddt  ttƒd ksJ ‚d S r}   )r   r:   r;   rD   rD   rD   rE   Útest_issue_7181f  s    r¡   c                      sŠ   t t d   ddt d   t d   tdt d  ƒ tt t t ƒ ttttdt d  ƒd ƒ ƒ ttƒ ‰ t	t
‡ fdd„ƒ d S )NrB   rZ   rK   c                      s    t ˆ ttdtffi dddœ¤ŽS )NrP   T)Z
as_meijergZneedeval)r   r;   r:   r   rD   ©ÚFrD   rE   Ú<lambda>v  s    ÿz!test_issue_8882.<locals>.<lambda>)r<   r;   r%   r"   r   r   r(   r&   r1   r9   r   rD   rD   r¢   rE   Útest_issue_8882j  s    
Jÿÿr¥   c                  C   s8   t ddd\} }tt| ƒ| |ƒtt| ƒ| |ƒks4J ‚d S )Nzx yTr   )r   r   r"   r   )r:   ÚyrD   rD   rE   Útest_issue_12591{  s    r§   N)grC   r   r   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   Zsympy.integrals.laplacer   r   r‰   r   r   Z
sympy.corer   Zsympy.core.numbersr   r   r   r   Zsympy.core.singletonr   Zsympy.core.symbolr   r   Z(sympy.functions.combinatorial.factorialsr   Z$sympy.functions.elementary.complexesr    r!   Z&sympy.functions.elementary.exponentialr"   r#   r$   ra   r%   rŒ   r&   r'   r(   r)   Zsympy.functions.special.besselr*   r+   r,   r-   Z'sympy.functions.special.delta_functionsr.   rv   r/   r0   Z'sympy.functions.special.gamma_functionsr1   Zsympy.functions.special.hyperr2   Zsympy.simplify.gammasimpr3   Zsympy.simplify.hyperexpandr4   Zsympy.simplify.trigsimpr5   Zsympy.testing.pytestr6   r7   r8   r9   Z	sympy.abcr:   r;   r<   r=   r>   r?   rb   rc   rd   rF   rG   rN   rV   rg   rh   rm   rx   rŽ   rš   r   rž   r    r¡   r¥   r§   rD   rD   rD   rE   Ú<module>   sX   L ?
`
!
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