o
    jgÊ“  ã                   @   sð  d dl 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mZmZ d dlmZ d dlmZmZmZ d dl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'm(Z(m)Z)m*Z* d dl+m,Z, d dl-m.Z. d dl/m0Z0m1Z1m2Z2 d dl3m4Z4 d dl5m6Z6m7Z7m8Z8 d dl9m:Z:m;Z; d dl<m=Z= d dl>m?Z? d dl@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZH d dlImJZJmKZKmLZLmMZM d dlNmOZOmPZP d dlQmRZRmSZS d dlTmUZU eOdd„ ƒZVeOdd„ ƒZWeOdd„ ƒZXeOd d!„ ƒZYd"S )#é    )Úlaplace_transformÚinverse_laplace_transformÚLaplaceTransformÚInverseLaplaceTransformÚ_laplace_deep_collectÚlaplace_correspondenceÚlaplace_initial_conds)ÚFunctionÚ
expand_mul)Ú
EulerGammaÚSubsÚ
DerivativeÚdiff)Úfactor_terms)ÚIÚooÚpi)ÚEq)ÚS)ÚSymbolÚsymbols©Úsimplify)ÚAbsÚre)ÚexpÚlogÚ	exp_polar)ÚcoshÚsinhÚcothÚasinh)Úsqrt)Ú	Piecewise)ÚatanÚcosÚsin)ÚAnd)Ú
lowergammaÚgammaÚ
uppergamma)Ú
DiracDeltaÚ	Heaviside)ÚSingularityFunction)Úlerchphi)ÚfresnelcÚfresnelsÚerfÚerfcÚEiÚCiÚexpintÚE1)ÚbesseliÚbesseljÚbesselkÚbessely)ÚslowÚwarns_deprecated_sympy)ÚMatrixÚeye)Úsc               
   C   s@=  t } t}tddd\}}}tdddd}tdƒ\}}}tdƒ}	tdƒ}
td	ƒ}td
ƒ}tdƒ}tt|| ||  ƒtdt|| ||  |d  ƒƒ |ƒt|| |d  |||   ƒtdt|| |||   ƒƒ ksnJ ‚t t||ƒ|dƒ|tdd}t	|||iƒ}t
|||g d¢iƒ}|td |tƒ dtd   dt  d ks¥J ‚dtt|ƒ|tƒ d  ¡ ddtd   ks½J ‚| || |d  |tdƒd   |tƒ|td  dtd   dttƒ dttdƒd     ddfksòJ ‚| |||  |tƒ| t| t ƒ t| t ƒ ddfksJ ‚| dt|| ƒ |tƒttƒtdt ƒ t|t ƒ tt|ƒttƒ ƒ ddfks?J ‚| t|ƒ||  |tƒt t|ƒ t|t ƒ tt|ƒttƒ ƒ ttƒtdt ƒ  ddfksrJ ‚| || tdƒ d  |tƒdttƒ ttƒ t|t ƒ tt|ƒttƒ ƒ dt|ƒ  ddfks§J ‚| |tdƒd  || d  |tƒt t|ƒ t|t ƒ tt|ƒttƒ ƒ ttƒtdt ƒ  ddfksàJ ‚| d|t|ƒ |d   |tƒtt|ƒ t|t ƒ tt|ƒttƒ ƒ ddfksJ ‚| || | |tƒt| d  t| t ƒ t|d |t ƒ ddfks3J ‚| |d ||  |tƒd|d  td|t ƒ ddfksRJ ‚| t|ƒ|tƒdtd  ddfksfJ ‚| td| ƒ|tƒdtd  ddfks|J ‚| t|| ƒ|tƒdt|  |dfks’J ‚| t|||  ƒ|tƒt| | ƒ| t  |dfks°J ‚| |t| | ƒ |tƒ|t d | dfksÊJ ‚| |t| ||  ƒ |tƒt|| ƒ|t d  | dfksìJ ‚| || t| | ƒ |tƒ||t d  | dfks
J ‚| t| t| ƒ ƒ|tƒtt|ƒ|t  ddfks'J ‚| t| t|ƒ ƒ|tƒ|t tt |ƒ ddfksDJ ‚| |tdƒd  td| ƒ ttdƒd ƒ |tƒtd tdƒ d  ddfksoJ ‚| |tdƒd  td| ƒ |tƒdttƒ dtd tdƒd    ddfks™J ‚| || t| | ƒ |tƒ|t | d  t|d ƒ | dfks¾J ‚| |t| |d  ƒ |tƒttƒ| ttd d|  ƒ ttdt|ƒ  ƒ dt|ƒ  ddfksóJ ‚| td|d  ƒ|tƒtdƒttƒ ttd d ƒ ttdƒt d ƒ d ddfks!J ‚| |td|d  ƒ |tƒ|ttd|d  ƒ|tƒ t dfksDJ ‚| |t| |d  ƒ |tƒdd|  tttdt|ƒ  ƒ dttƒ |tdƒd     ddfksyJ ‚| t| | ƒ|tƒdt|ƒ tdt ƒ tddt|ƒ ttƒ ƒ ddfks£J ‚| t|ƒt| | ƒ |tddttƒt|ƒttƒ dtdƒ   ttd ƒ tdt|ƒ ttƒ ƒ ddfksÞJ ‚| t| | ƒt|ƒ |tƒttƒtdt ƒ tdt|ƒ ttƒ ƒ ddfks	J ‚| t| | ƒ|t|ƒ  |tƒttƒtd| ƒ tdt|ƒ ttƒ ƒ ddfks6J ‚| tdt|| ƒ ƒ|tƒdt ttƒt|ƒ t|t ƒ tt|ƒtdt ƒ ƒ ttdƒd    ddfkspJ ‚| tdt|| ƒ ƒt|ƒ |tƒt|t ƒtt|ƒtdt ƒ ƒ ttƒtdt ƒ  ddfks¤J ‚| |d td| ƒ |tƒdtdƒ dt tdƒd   tddtdƒ ttƒ ƒ ddfksÕJ ‚| t|| ƒ|tƒ||d  td   |dfksðJ ‚| |t|| ƒd  |tƒd|d  | d |d  t td   d| dfksJ ‚| |t|| ƒd  |tddd|d  | td |d  td    d| dfksFJ ‚| td| ƒ|tƒtd td   ddfks^J ‚| t|| ƒ|tƒt|d  td   |dfksyJ ‚| t|| ƒd |tddd|d  td  td|d  td    d| dfks¥J ‚| t|d ƒ|tddttdƒ tdƒ td d  ddfksÇJ ‚| d!t|| | ƒd  |tƒ\}}}|d"td tt td| ƒ d| td| ƒ    d|d    ttd d|d       ¡ dksJ ‚| t|| ƒt|| ƒ |tdd|d |d  td   d| dfks6J ‚| t|| ƒ| |tƒt|t | t  ƒd |dfksUJ ‚| |tdƒ d  t|| ƒ |tƒttƒ t| t ƒt|t ƒ  |dfks€J ‚| tdt|| ƒ ƒ|tƒttƒt|ƒ t|t ƒ ttdƒd   ddfksªJ ‚| t|ƒtdt|| ƒ ƒ |tddt|ƒ ttdƒd   ttƒtd  d| t  t|t ƒ tt|ƒtdt ƒ ƒ d  ttd#ƒd   ddfksûJ ‚| tdt|| ƒ ƒt|ƒ |tƒttƒt|t ƒ tt|ƒtdt ƒ ƒ ttƒ ddfks-J ‚| tt|| ƒƒd t|ƒ |tƒttƒt|t ƒd  dttƒ  ddfksWJ ‚| |tdƒd  t|| ƒ |tƒ|t td$ƒ d  | t td$ƒ d   ttd$ƒd ƒ d |dfksJ ‚| tdt|| ƒ ƒ|tƒttƒt|ƒ t|t ƒ tt|ƒtdt ƒ ƒ ttdƒd   dt  ddfksÊJ ‚| t|ƒtdt|| ƒ ƒ |tƒttƒ|td   t|t ƒ ttdƒd   ddfksúJ ‚| tdt|| ƒ ƒt|ƒ |tƒttƒt|t ƒ ttƒ ddfk	s J ‚| tt|| ƒƒd t|ƒ |tƒttƒt|t ƒd  dttƒ  ddfk	sJJ ‚| t|ƒ|tddttƒ t t ddfk	scJ ‚| t|| ƒ |tddt|ƒttƒ t t ddfk	s‚J ‚| td||  ƒ|tƒtt| ƒ tt | ƒ t ddfk	s¤J ‚| t|| ƒ|tddtt|ƒ tt| ƒtt | ƒ  td  ddfk	sÍJ ‚| t|ƒt|ƒ |tddttƒttƒ tdƒ t  ttƒ ddfk	sôJ ‚| |tdƒd  t|ƒ |tddttƒd%ttƒ td&ƒ dt  d'  dttdƒd    ddfk
s*J ‚| |d t|ƒ |tddd(d)ttƒ tj tdƒd)   td    ¡ tjk
sUJ ‚| t|ƒd |tddttƒt d td d)  t ddfk
swJ ‚| t| | ƒt|ƒ |tddt|t ƒ t |t  | dfk
sœJ ‚| t|| ƒ|tƒ||d td   ddfk
s¶J ‚| t t|| ƒƒ|tƒ|t!tt d|  ƒ |d td   ddfk
sÜJ ‚| t|| ƒ| |tƒt"|t ƒddfk
sôJ ‚| t|| ƒd | |tƒtd|d  td  d ƒd ddfksJ ‚| t|| ƒd |d  |tƒ|t"d| t ƒ ttd|d  td  d ƒ d  ddfksJJ ‚| tdt|| ƒ ƒ|tƒttƒt|ƒ t| t ƒ ttdƒd   ddfksuJ ‚| tdt|| ƒ ƒ| |tƒttt|ƒtdt ƒ ƒ ddfks™J ‚| t|| ƒ|tƒt|d td   ddfks³J ‚| t|| ƒd |tƒd|d  td  td|d  td    ddfksÛJ ‚| t|ƒtdt|| ƒ ƒ |tddttƒ| td   t| t ƒ ttdƒd   ddfksJ ‚| tdt|| ƒ ƒt|ƒ |tƒttƒtdt ƒ t| t ƒ ddfks8J ‚| t|| ƒt|| ƒ |tƒd| | t td || d  td || d    ddfksjJ ‚| t|| ƒt|| ƒ |tƒ||d  |d  td   td || d  td || d    ddfks£J ‚| t|| ƒt|| ƒ |tƒt|d |d  td   td || d  td || d    ddfksÛJ ‚| | | t|| ƒ t|| ƒ |tddd|d  |d d|d  td   td   ddfksJ ‚| |t| | ƒ t|| ƒ |tƒ|| |d |t d   | dfks8J ‚| |t| | ƒ t|| ƒ |tƒ||t  |d |t d   | dfksbJ ‚| t|d ƒ|tdd\}}}|dksxJ ‚|ttdƒ tdƒ td d   dksJ ‚| t|| ƒ|tƒttd d|d   ƒttd|  ƒ t ddfksµJ ‚| tt|| ƒƒ|tƒt|ƒtt|t ƒ  ddfksÓJ ‚| t|| ƒtt|| ƒƒ |tddt|ƒ ttƒ|t   |dfksúJ ‚| tt|| ƒd ƒ|tdddt tt|ƒ ttƒ ƒt  ddfks!J ‚| tt|| ƒƒ|tddt|ƒ tt|t ƒ  dt  | dfksGJ ‚| t|| ƒtt|| ƒƒ |tƒdt|ƒttƒ t  ddfkskJ ‚| tt|| ƒd ƒ|tƒtt|ƒ ttƒ ƒt ddfksŒJ ‚| td|| ƒ|tƒdt|d td  ƒ ddfks©J ‚| td|| ƒ|tdd||d td  tt|d td  ƒ   ddfksÒJ ‚| td|| ƒ|tdd|d t|d td  ƒtt|d td  ƒ d   ddfksJ ‚| |td|| ƒ |tƒt|d td  tdƒd   ddfks$J ‚| |td|| ƒ |tƒ||d td  tdƒd   ddfksGJ ‚| |d td|| ƒ |tƒd|d  |d td  tdƒd   ddfkspJ ‚| tddt|| ƒ ƒ|tƒt| t ƒt ddfksŽJ ‚| |tdƒd  tddt|| ƒ ƒ |tƒ|tdƒd  t| t ƒ td  ddfks¾J ‚| td|t|d ||  ƒ ƒ|tddt|tt|d td  ƒ  ƒt|d td  ƒ ddfksóJ ‚| t#d|| ƒ|tƒdt|d  td  ƒ |dfksJ ‚| t#d|| ƒ|tdd||d  td  tt|d  td  ƒ   |dfks<J ‚| t#d|| ƒ|tdd|d t|d  td  ƒtt|d  td  ƒ d   |dfksmJ ‚| |t#d|| ƒ |tƒt|d  td  tdƒd   |dfks‘J ‚| |t#d|| ƒ |tƒ||d  td  tdƒd   |dfksµJ ‚| |d t#d|| ƒ |tƒd|d  |d  td  tdƒd   |dfksßJ ‚| |tdƒd  t#ddt|| ƒ ƒ |tƒ|tdƒd  t|t ƒ td  ddfksJ ‚| t$d|| ƒ|tƒdt%t| ƒ tt|d td  ƒ  ddfks3J ‚| td|| ƒ|tƒttt|d  td  ƒ | ƒt|d  td  ƒ | dfksaJ ‚| t|| ƒd |tddd*|d  td+|d  d,|d  td   td    ddfks‘J ‚| t&|d ƒt|d ƒ |tƒttt ƒ td d  ddfks´J ‚| ||	|ƒ ||ƒ|t|	|ƒ||ƒ t dfksÏJ ‚| |t&|d ƒ |	|d ƒ |tƒ|t|	|d ƒ|tƒ t dfksôJ ‚| |t&|d ƒ |	|d ƒ |tƒ|t|	|ƒ|tƒ tt ƒ t dfksJ ‚| ||	|| ƒ |tƒ|| t|	|ƒ||t ƒ t dfks=J ‚| t|	|ƒ | ƒ|tƒdt|	|ƒ  t'|	|ƒƒ dfks]J ‚| t| | ƒ|	|ƒ |tƒt|	|ƒ||t ƒt dfks}J ‚| t| | ƒtt|| ƒd ƒ |tƒtt|ƒ t|t ƒ ƒ|t  | dfksªJ ‚| t|| ƒ|	|ƒ |tƒt|	|ƒ|| t ƒd t|	|ƒ||t ƒd  t dfksØJ ‚| t|| ƒ| |tddd| t |d d|d  td   td   |dfksJ ‚| t|| ƒ|	|ƒ |tƒt|	|ƒ|| t ƒd t|	|ƒ||t ƒd  t dfks2J ‚| t|| ƒ| |tdddd|t d   dd|t d    |dfksZJ ‚| t|| ƒ|	|ƒ |tddt(t|	|ƒ|t( | t ƒ t|	|ƒ|t(| t ƒ  d t dfksJ ‚| t|	|ƒƒ|tƒtt|	|ƒƒ|tƒt dfksªJ ‚| t|| ƒ| |tddd| t |d d|d  td   td   ddfksÖJ ‚| t|| ƒ|	|ƒ |tƒt|	|ƒ|t( | t ƒd t|	|ƒ|t(| t ƒd  t dfksJ ‚| t|| ƒ| |tdd|d  td  |d d|d  td   td   ddfks7J ‚| t|| | ƒd |	|ƒ |tƒ\}}}|t ksTJ ‚| t|	|ƒ|tƒd t|	|ƒ|dt( | t ƒtdt( | ƒ d  t|	|ƒ|dt( | t ƒtdt( | ƒ d   dks–J ‚| t|| | ƒd |	|ƒ |tdd}t	||	|
iƒ|
tƒd |
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J ‚| t&|ƒ|tƒdt ddfks2J ‚| t&|| ƒ|tƒt| t ƒt ddfksKJ ‚| t&|d ƒ|tƒtt ƒt ddfksbJ ‚| t&d| d ƒ|tƒtdt ƒt ddfks|J ‚| t&d| d ƒ|tƒdt ddfks’J ‚| t&d| d ƒ|tdddt tdt ƒt  ddfks²J ‚| ||ƒt&|| ƒ |tƒt||ƒt&|| ƒ |tƒt dfksÕJ ‚| t&|| ƒ||ƒ |tƒt||| ƒ|tƒt| t ƒ t dfksûJ ‚| t&|| ƒ||ƒ |tƒt||ƒ|tƒt dfksJ ‚| t&| | ƒ||ƒ |tƒt||ƒ|tƒt||| ƒ|tƒt| t ƒ  t dfksGJ ‚| t&| | ƒ||ƒ |tƒd2ks[J ‚t t2|ƒ|tddttd dt  ƒ t2tt ƒ tdƒttd dt  td  ƒ d  ttd dt  ƒt3tt ƒ  t ddfks¢J ‚t t3|ƒ|tddttd dt  ƒt3tt ƒ ttd dt  ƒt2tt ƒ  tdƒttd dt  td  ƒ d  t ddfksèJ ‚t4t|ƒ|t| ƒ g|t| ƒ t|ƒggƒ}t4dtd  td d gtd d dtd  ggƒ}t5ƒ . t4dtd  ddftd d ddfgtd d ddfdtd  ddfggƒ}W d   ƒ n	1 sOw   Y  t5ƒ  | ||tƒ|kscJ ‚W d   ƒ n	1 snw   Y  | ||td3d4|ddfksƒJ ‚| ||tdd|ksJ ‚| ||td3dd5|ksžJ ‚d S )6Nza, 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„}‡‡fdd„}‡ ‡‡‡fdd„}|t| ˆ ƒˆ ˆƒ ƒˆ| ˆ ƒksLJ ‚|| ˆ ˆ| ƒ ƒ| ˆˆƒ t| ˆ ƒ kseJ ‚|tˆ ˆƒˆdƒƒˆd  ˆˆƒ kszJ ‚|tˆ ˆƒˆdƒƒˆd ˆˆƒ ksŽJ ‚|dƒdtˆƒ ksšJ ‚|dˆ ƒtˆƒks¦J ‚|dˆd  ƒˆtˆƒ ks¶J ‚|dˆd  ƒˆd tˆƒ d ksÊJ ‚|dˆ|  ƒˆ|d  tˆƒ t|ƒ ksâJ ‚|||ˆ  ƒ|t| ˆ ƒ tˆƒ ksùJ ‚|ˆ|ˆ  ƒ| t| ˆ ƒ tˆƒ tˆƒ ksJ ‚||ˆ ˆ| d  ƒ|| ˆ t| ˆ ƒ tˆƒ t| ˆ ƒtˆƒ   ksBJ ‚||ˆ| ˆ|   ƒ|t|ˆ ƒt|ˆ ƒ  tˆ ||  ƒ tˆƒ ||  kspJ ‚||ˆ ˆ| ˆ|   ƒ||t|ˆ ƒ |t|ˆ ƒ   tˆ ||  ƒ tˆƒ ||  ks¤J ‚|ˆ|ˆ d  ƒˆ| ˆ d  t| ˆ ƒ tˆƒ d ksÇJ ‚|dˆ|ˆ d   ƒ|d  ˆd  d| ˆ  dt|ˆ ƒ  d t| ˆ ƒ tˆƒ d|d   ks J ‚|dˆ|ˆ |   ƒtˆƒt||ˆ ƒ || t|ƒ  ks"J ‚|ˆ| |  ƒˆ|d  t|ˆ ƒ tˆƒ t|ƒ ksBJ ‚||ˆ d ƒˆt| ˆ ƒ tˆƒ ksZJ ‚||ˆ d ƒˆd t| ˆ ƒ tˆƒ d ksvJ ‚|ˆd ˆd d  ƒtˆƒ tˆƒ tˆƒ ks’J ‚|ddˆd d   ƒtˆƒ tˆƒ tˆƒ ks®J ‚|||d ˆd   ƒt|ˆ ƒtˆƒ ksÇJ ‚|ˆˆd |d   ƒt	|ˆ ƒtˆƒ ksàJ ‚|||d |ˆ d   ƒt| ˆ ƒt|ˆ ƒ tˆƒ ksJ ‚||ˆ |d |ˆ d   ƒ|| t| ˆ ƒ t|ˆ ƒ | t| ˆ ƒt	|ˆ ƒ   tˆƒ ks:J ‚|dˆd ˆd d   ƒˆtˆƒ tˆƒtˆƒ  ksYJ ‚||ˆ |d ˆ| d  |d   ƒ|| | t|ˆ | ƒ |t	|ˆ | ƒ   t| ˆ ƒ tˆƒ ||d   ksšJ ‚||ˆd  | |d ˆd  d  ƒd|d  | t|ˆ ƒ |d | | |ˆ t	|ˆ ƒ t|ˆ ƒ   tˆƒ d|d   ksáJ ‚||ˆd |d   ƒ|t
|ˆ ƒ tˆƒ | ksþJ ‚||ˆd |d   ƒ|t|ˆ ƒtˆƒ d|  t| ˆ ƒtˆƒ d|    ks,J ‚||ˆ ˆd |d   ƒ|t|ˆ ƒ tˆƒ ksIJ ‚||ˆˆ|   ƒ|tˆƒ| t| ˆ ƒtˆƒ |   kskJ ‚|ˆd ˆd ˆd dˆ  d   ƒdtdˆ ƒ dt	dˆ ƒ  tˆƒ d tdˆ ƒ tˆƒ d ks¤J ‚|d|ˆd  |ˆ  |  ƒdt| ˆ d|  ƒ tˆtd| | |d  ƒ d|  ƒ tˆƒ td| | |d  ƒ ksèJ ‚|dtˆƒ |ˆ |  ƒt|ˆ | ƒtˆƒ tt|ƒtˆƒ t|ƒ ƒ t|ƒt|ƒ  ksJ ‚|dt| ˆ ƒ |ˆ |  ˆ ƒd| tˆƒ ttƒ|  |tdƒd  t|ˆ | ƒ tt|ƒtˆƒ t|ƒ ƒ |tdƒd    tˆƒ |t| ƒ  ksnJ ‚|dt|ˆ ƒ|  ƒ| t|d ˆ | ƒ t|tˆƒ t|ƒ ƒ | dttƒt|ƒ tˆƒ   tˆƒ ksªJ ‚||ˆ |tˆƒ |  ƒt|d ˆ |d  ƒ t|tˆƒ | ƒ d tˆƒ ksØJ ‚||| tˆƒ tˆƒt|ƒ  ˆ|  ƒt|ƒt|ƒ t|ˆ ƒ tt|ƒtˆƒ ƒ |t|ˆ ƒ tt|ƒtˆƒ ƒ  |t|ˆ ƒ  tˆƒ ks'J ‚|dtˆƒ t|ˆ ƒ|  ƒt|d ˆ | ƒtˆƒ t|tˆƒ t|ƒ ƒ t|ƒ ksXJ ‚||d tˆƒ|  ˆtdƒd   ƒd| tˆƒ ttƒ t|d ˆ ƒt|tˆƒ ƒ  d tˆƒ ks‘J ‚||| t|ƒ ˆ|  tˆƒ tˆƒt|ƒ  ƒt|ƒt|ˆ ƒ tt|ƒtˆƒ ƒ t|ƒt|ˆ ƒ tt|ƒtˆƒ ƒ  t|ƒt|ˆ ƒ  tˆƒ ksäJ ‚|dtˆƒt|ƒ d  ƒdt|ƒ tˆƒ ttƒ d| ˆ d tt|ƒtˆƒ ƒd  t|ˆ ƒ  d tˆƒ ks#J ‚|dtˆƒt|ƒ d  ˆ ƒdˆ d|  t|ˆ ƒ tt|ƒtˆƒ ƒ d|  dtˆƒ ttƒt|ƒ   tˆƒ ksdJ ‚|dtˆƒ| d  tˆƒ ƒd| ˆ t|d ˆ ƒ t|tˆƒ ƒ dtˆƒ ttƒ  tˆƒ ks›J ‚|dtˆƒ| d  ƒ| ˆ d|d  ˆ d  t|d ˆ ƒ t|tˆƒ ƒ dtˆƒ |d ˆ d  ttƒ  tˆƒ ksßJ ‚|tˆƒtˆƒ| d  ƒ dttƒ|d  ˆ dttƒ t|tˆƒ ƒ dt|d  ˆ ƒ |tˆƒ    |d  ˆd  d|d  ˆ d  tj  t|d ˆ ƒ d ttƒ| tˆƒ |d ˆ d  d   tˆƒ t|d  ˆ    ¡ }	|	dksbJ ‚|dtˆƒ tˆƒ| d  ƒ tˆƒtˆƒd|d  ˆ d ttƒ| tˆƒ t|d ˆ ƒ t|tˆƒ ƒ d  d  ttƒ|     ¡ }	|	dks²J ‚t|dtˆƒ| d  ƒƒdd|d  ˆtdƒd   d|d  ˆ d  dttƒ  ˆd|d  ˆd  d|d  ˆ  d  t|d ˆ ƒ t|tˆƒ ƒ d   tˆƒ k	sJ ‚|tˆƒ| ˆtˆƒ|   ƒdt|d ˆ ƒ t|tˆƒ ƒ d tˆƒ k	s:J ‚|tˆƒ| d ˆtˆƒ| d   ƒdd|d  ˆ t|d ˆ ƒ t|tˆƒ ƒ  dttƒ | tˆƒ  tˆƒ k	s{J ‚|tˆƒ| d ˆtˆƒ| d   ƒtˆƒdd|d  ˆd  d|d  ˆ  d  t|d ˆ ƒ t|tˆƒ ƒ dttƒ | tˆƒ d|d  ˆ d   d  k	sÔJ ‚|tˆ| ƒˆ|  ƒtˆƒt| ˆ ƒtˆƒ ttƒ t|| ƒt| ˆ ƒ tt|| ƒtˆƒ ƒ   k
sJ ‚|dtˆ| ƒ ˆ|  ƒtˆƒdt|| ƒ t| ˆ ƒ tt|| ƒtˆƒ ƒ  k
s?J ‚|dtˆd |d  ƒ ƒtˆƒtd|ˆ ƒ k
s[J ‚|dtˆd |d  ƒ ƒtˆƒtd|ˆ ƒ k
swJ ‚|ˆd tdˆ ƒ dtˆƒ  d tdˆ ƒ ˆˆd d   ƒ t	ˆƒtˆƒ  dt	ˆd ƒ tˆd ƒ  dt	ˆd ƒ tˆd ƒ  dtˆd ƒ  dtˆd ƒ   ¡ dk
sÕJ ‚d S )Nr?   zk, n, tT©Úrealz
a, b, c, dr@   rD   rE   c                    s   t | ˆ ˆƒS ©N©r   ©rF   ©r?   ro   r~   r   rk   ñ  s   z+test_inverse_laplace_transform.<locals>.ILTc                    s   t | ˆ ˆddS )NTr   r„   r…   r†   r~   r   ÚILTSô  s   z,test_inverse_laplace_transform.<locals>.ILTSc                    s   t t| ˆˆƒˆˆ iƒS rƒ   )r   r   r…   ©rE   rD   r?   ro   r~   r   ÚILTF÷  s   ÿz,test_inverse_laplace_transform.<locals>.ILTFrJ   rM   rI   rQ   rS   rb   rV   rX   rN   r_   r^   rT   r   é   )r   r	   r   r   r+   r,   r)   r(   r&   r%   r   r   r"   r1   r   r   r2   ÚHalfr   r   r8   r7   )
ÚkÚnrl   rm   rn   Údrk   r‡   r‰   rq   r~   rˆ   r   Útest_inverse_laplace_transformé  sø  ,2*( (0.:8
ÿ<
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ÿ$
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ÿþ
þ:<
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þ
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ÿÿþþ
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ÿ. ÿ*þþþ&ýüüÿú "ÿÿÿþ
ýÿû<:ÿÿÿþ
ÿ.
ÿ&2ÿþ,:ÿ*ÿÿÿ0ÿÿ 4ÿ ÿ ÿDÿÿþþýür   c                  C   s²  ddl m}  t}tddd\}}}}tddd\}}tdƒ\}}	td	ƒ}
td
ƒ}dd„ }||tƒt|ƒ}t||
|iƒ|
|ƒksAJ ‚|t| t ƒt t|ƒt| | ƒksWJ ‚|t| t ƒ|t  t|ƒt| | |  ƒt| | ƒ ksyJ ‚||t |d |t d   t|ƒt| | ƒt	|| ƒ t|ƒ ksžJ ‚|t| t ƒt|  t|ƒ| | |d  t| | ƒ t
|ƒ ksÃJ ‚|t| t ƒttd d ƒ t|ƒt| | ƒtd|| ƒ ksæJ ‚|dtttd ƒ  t|ƒt|ƒtt|ƒƒ ksJ ‚|t| t ƒt|  t|ƒ|| |d  t|| ƒ t
|ƒ ks%J ‚|t| t ƒtdtd  ƒ t|ƒt|| ƒtd|| ƒ ksHJ ‚t||| tttd |d  ƒ |   ttd |d  ƒ t|ƒ t¡ƒt|ƒt||| ƒ ks|J ‚||| tttd |d  ƒ |   ttd |d  ƒ t|dd t¡t|ƒt||| ƒ ks°J ‚|dtttd ƒ  t|ƒt|ƒtt|ƒƒ ksËJ ‚|ttdƒ tddgddggƒ  ¡ t|ƒtt|ƒt|ƒ dgdtd| ƒt|ƒ ggƒksüJ ‚|td |
tƒ t|ƒtt|
tƒt|d ƒ|dfƒksJ ‚|t	tƒt|ƒtt	tƒt|d ƒks-J ‚|dtdt ƒ dtdt ƒ  t|ƒdttdt ƒt|d ƒ d| |d ƒ  ksYJ ‚t	tdƒd ƒ}||tdt ƒ t|ƒ|| |d ƒ ksxJ ‚|tdt ƒt|ƒttdt ƒt|d ƒksJ ‚tddd}|t|t ƒt|ƒtt|t ƒt|d ƒks®J ‚|td |d td   t|ƒ| t|| ƒ t|ƒ | |ƒ ksÒJ ‚|td |
tƒd|d td     t|ƒ| t|| ƒ t|ƒ | |ƒ tt|
tƒt|d ƒ|dfƒ ks
J ‚|t|t ƒt|ddtt|t ƒt|d ƒdfks&J ‚tdddtddddfD ]#}	|t|	t ƒt|dd}
t|
tƒrK|
d n|
}
|
j| ksUJ ‚q3d S )Nr   )r+   za b c dTr@   zn, rr   zt zrD   rE   c                 S   s   t t| ƒƒ t¡S rƒ   )r   r
   Úrewriter&   )Úexprr~   r~   r   Úsimp_hypª  s   z4test_inverse_laplace_transform_old.<locals>.simp_hyprI   rQ   r   r\   rJ   rS   iùÿÿÿrU   rZ   ÚrFrK   Úz)Úextended_real)Ú	imaginaryÚzero)Ú'sympy.functions.special.delta_functionsr+   r   r   r	   r?   r   r   r,   r%   r)   r"   r8   r1   r   r   r7   r>   r=   Úinvr   r   r&   r   Ú
isinstanceÚtupleÚfunc)r+   rk   rl   rm   rn   rŽ   r   r“   ro   r”   rD   rE   r’   rr   r~   r~   r   Ú"test_inverse_laplace_transform_old   sš   ,D" ÿ&ÿ$ÿ6"
ÿ$
ÿ<ÿÿ
þ:ÿ
þ6*.
ÿ
ÿ(&ÿ
ÿ.00 
ÿ* ÿ
ÿ
ÿ
ÿür   c                  C   sÂ  t dƒ} t dƒ}t ddd}tt| ƒ| tƒtdtd  ƒ d t ddfks(J ‚tt|| ƒ| tdd	ttttt	 ƒ d|ƒdt
|ƒtjkfksIJ ‚ttd| ƒ| tdd	ttd ƒt ddfksaJ ‚ttd| ƒ| tdd	tttd ƒ td  ddfks}J ‚ttdtd  ƒ d t t|ƒ ¡ t|ƒt|ƒ ksšJ ‚tttd ƒt t| dd	 t¡t| ƒt| ƒ ksµJ ‚ttttd ƒ td  t| dd	 t¡ ¡ td| ƒt| ƒ  t¡ t¡ ¡ ksßJ ‚d S )
Nrq   rl   ÚuT)ÚpolarrQ   rI   r   r   )r   r   r4   r?   r   r5   r.   r   r   r   r   r   rh   r   ri   r,   r   r6   r3   )rq   rl   rž   r~   r~   r   Útest_expinté  s:   4&ÿÿÿ$ÿÿþþ
 ýr    N)ZÚsympy.integrals.laplacer   r   r   r   r   r   r   Úsympy.core.functionr	   r
   Ú
sympy.corer   r   r   r   Úsympy.core.exprtoolsr   Úsympy.core.numbersr   r   r   Úsympy.core.relationalr   Úsympy.core.singletonr   Úsympy.core.symbolr   r   Úsympy.simplify.simplifyr   Ú$sympy.functions.elementary.complexesr   r   Ú&sympy.functions.elementary.exponentialr   r   r   Ú%sympy.functions.elementary.hyperbolicr   r   r    r!   Ú(sympy.functions.elementary.miscellaneousr"   Ú$sympy.functions.elementary.piecewiser#   Ú(sympy.functions.elementary.trigonometricr$   r%   r&   Úsympy.logic.boolalgr'   Ú'sympy.functions.special.gamma_functionsr(   r)   r*   r˜   r+   r,   Ú-sympy.functions.special.singularity_functionsr-   Ú&sympy.functions.special.zeta_functionsr.   Ú'sympy.functions.special.error_functionsr/   r0   r1   r2   r3   r4   r5   r6   Úsympy.functions.special.besselr7   r8   r9   r:   Úsympy.testing.pytestr;   r<   Úsympy.matricesr=   r>   Ú	sympy.abcr?   r€   r   r   r    r~   r~   r~   r   Ú<module>   sJ   $ (
   I
 7
H