U
    L?hÊ“  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sÜt‚t t||ƒ|dƒ|t	dd}t
|||iƒ}t|||dddddgiƒ}|t	d |t	ƒ dt	d   dt	  d ksRt‚dtt|ƒ|t	ƒ d  ¡ ddt	d   ks„t‚| || |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sðt‚| |||  |t	ƒ| t| t	 ƒ t| t	 ƒ ddfks2t‚| 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sŠt‚| 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sðt‚| || 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sZt‚| |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sÌt‚| d|t|ƒ |d   |t	ƒtt|ƒ t|t	 ƒ tt|ƒtt	ƒ ƒ ddfks$t‚| || | |t	ƒt	| d  t| t	 ƒ t|d |t	 ƒ ddfksrt‚| |d ||  |t	ƒd|d  td|t	 ƒ ddfks°t‚| t|ƒ|t	ƒdt	d  ddfksØt‚| td| ƒ|t	ƒdt	d  ddfkst‚| t|| ƒ|t	ƒdt	|  |dfks0t‚| t|||  ƒ|t	ƒt| | ƒ| t	  |dfkslt‚| |t| | ƒ |t	ƒ|t	 d | dfks t‚| |t| ||  ƒ |t	ƒt|| ƒ|t	 d  | dfksät‚| || t| | ƒ |t	ƒ||t	 d  | dfks t‚| t| t| ƒ ƒ|t	ƒtt	|ƒ|t	  ddfksZt‚| t| t|ƒ ƒ|t	ƒ|t	 tt	 |ƒ ddfks”t‚| |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sêt‚| |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>t‚| || t| | ƒ |t	ƒ|t	 | d  t|d ƒ | dfksˆt‚| |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sòt‚| 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sNt‚| |td|d  ƒ |t	ƒ|ttd|d  ƒ|t	ƒ t dfks”t‚| |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sþt‚| t| | ƒ|t	ƒdt|ƒ tdt	 ƒ tddt|ƒ tt	ƒ ƒ ddfk	sRt‚| 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	sÈt‚| 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t‚| t| | ƒ|t|ƒ  |t	ƒttƒtd| ƒ tdt|ƒ tt	ƒ ƒ ddfk
sxt‚| 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
sìt‚| 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sTt‚| |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s¶t‚| t|| ƒ|t	ƒ||d  t	d   |dfksìt‚| |t|| ƒd  |t	ƒd|d  | d!|d  t	 t	d   d| dfks@t‚| |t|| ƒd  |t	ddd|d  | t	d!|d  t	d    d| dfks˜t‚| td| ƒ|t	ƒt	d!t	d   ddfksÈt‚| t|| ƒ|t	ƒt	|d  t	d   |dfksþt‚| t|| ƒd |t	ddd|d  t	d  t	d|d  t	d    d| dfksVt‚| t|d ƒ|t	ddt	tdƒ tdƒ t	d d  ddfksšt‚| 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,t‚| t|| ƒt|| ƒ |t	dd|d!|d  t	d   d| dfksxt‚| t|| ƒ| |t	ƒt|t	 | t	  ƒd |dfks¶t‚| |tdƒ d  t|| ƒ |t	ƒttƒ t| t	 ƒt|t	 ƒ  |dfkst‚| tdt|| ƒ ƒ|t	ƒttƒt|ƒ t|t	 ƒ t	tdƒd   ddfks`t‚| 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st‚| 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sft‚| tt|| ƒƒd t|ƒ |t	ƒttƒt|t	 ƒd  dtt	ƒ  ddfksºt‚| |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s,t‚| 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s t‚| 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s t‚| tdt|| ƒ ƒt|ƒ |t	ƒttƒt|t	 ƒ tt	ƒ ddfksLt‚| tt|| ƒƒd t|ƒ |t	ƒttƒt|t	 ƒd  dtt	ƒ  ddfks t‚| t|ƒ|t	ddtt	ƒ t t	 ddfksÒt‚| t|| ƒ |t	ddt|ƒtt	ƒ t t	 ddfkst‚| td||  ƒ|t	ƒtt	| ƒ tt	 | ƒ t	 ddfksTt‚| t|| ƒ|t	ddt	t|ƒ tt	| ƒtt	 | ƒ  t	d  ddfks¦t‚| t|ƒt|ƒ |t	ddttƒtt	ƒ tdƒ t  tt	ƒ ddfksôt‚| |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`t‚| |d t|ƒ |t	ddd)d*tt	ƒ tj tdƒd*   t	d    ¡ tj ks¶t‚| t|ƒd |t	ddtt	ƒt d td d*  t	 ddfksút‚| t| | ƒt|ƒ |t	ddt|t	 ƒ t |t	  | dfksDt‚| t|| ƒ|t	ƒ||d t	d   ddfksxt‚| t!t|| ƒƒ|t	ƒ|t"tt	 d|  ƒ |d t	d   ddfksÄt‚| t|| ƒ| |t	ƒt#|t	 ƒddfksôt‚| t|| ƒd | |t	ƒtd|d  t	d  d ƒd ddfks<t‚| t|| ƒd |d  |t	ƒ|t#d| t	 ƒ t	td|d  t	d  d ƒ d  ddfks t‚| tdt|| ƒ ƒ|t	ƒttƒt|ƒ t| t	 ƒ t	tdƒd   ddfksöt‚| tdt|| ƒ ƒ| |t	ƒttt|ƒtdt	 ƒ ƒ ddfks>t‚| t|| ƒ|t	ƒt	|d t	d   ddfksrt‚| t|| ƒd |t	ƒd|d  t	d  t	d|d  t	d    ddfksÂt‚| 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*t‚| tdt|| ƒ ƒt|ƒ |t	ƒttƒtdt	 ƒ t| t	 ƒ ddfks|t‚| t|| ƒt|| ƒ |t	ƒd| | t	 t	d || d  t	d || d    ddfksàt‚| t|| ƒt|| ƒ |t	ƒ||d  |d  t	d   t	d || d  t	d || d    ddfksRt‚| t|| ƒt|| ƒ |t	ƒt	|d |d  t	d   t	d || d  t	d || d    ddfksÂt‚| | | t|| ƒ t|| ƒ |t	ddd|d  |d d|d  t	d   t	d   ddfks,t‚| |t| | ƒ t|| ƒ |t	ƒ|| |d |t	 d   | dfks|t‚| |t| | ƒ t|| ƒ |t	ƒ||t	  |d |t	 d   | dfksÐt‚| t|d ƒ|t	dd\}}}|dksüt‚|t	tdƒ tdƒ t	d d   dks*t‚| t|| ƒ|t	ƒtt	d d|d   ƒtt	d|  ƒ t	 ddfksvt‚| tt|| ƒƒ|t	ƒt|ƒt	t|t	 ƒ  ddfks²t‚| t|| ƒtt|| ƒƒ |t	ddt|ƒ tt	ƒ|t	   |dfks t‚| tt|| ƒd ƒ|t	dddt	 tt|ƒ tt	ƒ ƒt	  ddfksNt‚| tt|| ƒƒ|t	ddt|ƒ t	t|t	 ƒ  dt	  | dfksšt‚| t|| ƒtt|| ƒƒ |t	ƒdt|ƒtt	ƒ t	  ddfksât‚| tt|| ƒd ƒ|t	ƒtt|ƒ tt	ƒ ƒt	 ddfks$t‚| td|| ƒ|t	ƒdt|d t	d  ƒ ddfks^t‚| td|| ƒ|t	dd||d t	d  t	t|d t	d  ƒ   ddfks°t‚| td|| ƒ|t	dd|d t|d t	d  ƒt	t|d t	d  ƒ d   ddfkst‚| |td|| ƒ |t	ƒt	|d t	d  tdƒd   ddfksTt‚| |td|| ƒ |t	ƒ||d t	d  tdƒd   ddfksšt‚| |d td|| ƒ |t	ƒd|d  |d t	d  tdƒd   ddfksìt‚| tddt|| ƒ ƒ|t	ƒt| t	 ƒt	 ddfks(t‚| |tdƒd  tddt|| ƒ ƒ |t	ƒ|tdƒd  t| t	 ƒ t	d  ddfksˆt‚| td|t|d ||  ƒ ƒ|t	ddt|t	t|d t	d  ƒ  ƒt|d t	d  ƒ ddfksòt‚| t$d|| ƒ|t	ƒdt|d  t	d  ƒ |dfk s.t‚| t$d|| ƒ|t	dd||d  t	d  t	t|d  t	d  ƒ   |dfk s„t‚| t$d|| ƒ|t	dd|d t|d  t	d  ƒt	t|d  t	d  ƒ d   |dfk sæt‚| |t$d|| ƒ |t	ƒt	|d  t	d  tdƒd   |dfk!s.t‚| |t$d|| ƒ |t	ƒ||d  t	d  tdƒd   |dfk!svt‚| |d t$d|| ƒ |t	ƒd|d  |d  t	d  tdƒd   |dfk!sÊt‚| |tdƒd  t$ddt|| ƒ ƒ |t	ƒ|tdƒd  t|t	 ƒ t	d  ddfk"s(t‚| t%d|| ƒ|t	ƒdt&t	| ƒ tt|d t	d  ƒ  ddfk"srt‚| td|| ƒ|t	ƒtt	t|d  t	d  ƒ | ƒt|d  t	d  ƒ | dfk"sÎt‚| t|| ƒd |t	ddd+|d  t	d,|d  d-|d  t	d   t	d    ddfk#s.t‚| t'|d ƒt|d ƒ |t	ƒt	tt	 ƒ t	d d  ddfk#stt‚| ||	|ƒ ||ƒ|t|	|ƒ||ƒ t dfk#sªt‚| |t'|d ƒ |	|d ƒ |t	ƒ|t|	|d ƒ|t	ƒ t dfk#sôt‚| |t'|d ƒ |	|d ƒ |t	ƒ|t|	|ƒ|t	ƒ tt	 ƒ t dfk$sDt‚| ||	|| ƒ |t	ƒ|| t|	|ƒ||t	 ƒ t dfk$s†t‚| t|	|ƒ | ƒ|t	ƒdt	|	|ƒ  t(|	|ƒƒ dfk$sÆt‚| t| | ƒ|	|ƒ |t	ƒt|	|ƒ||t	 ƒt dfk%st‚| t| | ƒtt|| ƒd ƒ |t	ƒtt|ƒ t|t	 ƒ ƒ|t	  | dfk%s`t‚| t|| ƒ|	|ƒ |t	ƒt|	|ƒ|| t	 ƒd t|	|ƒ||t	 ƒd  t dfk%s¼t‚| t|| ƒ| |t	ddd| t	 |d d|d  t	d   t	d   |dfk&st‚| t|| ƒ|	|ƒ |t	ƒt|	|ƒ|| t	 ƒd t|	|ƒ||t	 ƒd  t dfk&spt‚| t|| ƒ| |t	dddd|t	 d   dd|t	 d    |dfk&sÀt‚| t|| ƒ|	|ƒ |t	ddt)t|	|ƒ|t) | t	 ƒ t|	|ƒ|t)| t	 ƒ  d t dfk's*t‚| t|	|ƒƒ|t	ƒtt|	|ƒƒ|t	ƒt dfk's`t‚| t|| ƒ| |t	ddd| t	 |d d|d  t	d   t	d   ddfk's¸t‚| t|| ƒ|	|ƒ |t	ƒt|	|ƒ|t) | t	 ƒd t|	|ƒ|t)| t	 ƒd  t dfk(st‚| t|| ƒ| |t	dd|d  t	d  |d d|d  t	d   t	d   ddfk(szt‚| t|| | ƒd |	|ƒ |t	ƒ\}}}|t k(s´t‚| 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)s8t‚| t|| | ƒd |	|ƒ |t	dd}t
||	|
iƒ|
t	ƒd |
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   	      s¼  t dƒ‰t ddd\} }‰t ddd\}}}}tdƒ‰tdƒ‰ ‡‡fd	d
„}‡‡fdd„}‡ ‡‡‡fdd„}|t| ˆ ƒˆ ˆƒ ƒˆ| ˆ ƒks˜t‚|| ˆ ˆ| ƒ ƒ| ˆˆƒ t| ˆ ƒ ksÊt‚|tˆ ˆƒˆdƒƒˆd  ˆˆƒ ksôt‚|tˆ ˆƒˆdƒƒˆd ˆˆƒ kst‚|dƒdtˆƒ ks8t‚|dˆ ƒtˆƒksRt‚|dˆd  ƒˆtˆƒ kstt‚|dˆd  ƒˆd tˆƒ d ksžt‚|dˆ|  ƒˆ|d  tˆƒ t|ƒ ksÐt‚|||ˆ  ƒ|t| ˆ ƒ tˆƒ ks t‚|ˆ|ˆ  ƒ| t| ˆ ƒ tˆƒ tˆƒ ks:t‚||ˆ ˆ| d  ƒ|| ˆ t| ˆ ƒ tˆƒ t| ˆ ƒtˆƒ   ks’t‚||ˆ| ˆ|   ƒ|t|ˆ ƒt|ˆ ƒ  tˆ ||  ƒ tˆƒ ||  ksît‚||ˆ ˆ| ˆ|   ƒ||t|ˆ ƒ |t|ˆ ƒ   tˆ ||  ƒ tˆƒ ||  ksVt‚|ˆ|ˆ d  ƒˆ| ˆ d  t| ˆ ƒ tˆƒ d ksœt‚|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t‚|dˆ|ˆ |   ƒtˆƒt||ˆ ƒ || t|ƒ  ksRt‚|ˆ| |  ƒˆ|d  t|ˆ ƒ tˆƒ t|ƒ ks’t‚||ˆ d ƒˆt| ˆ ƒ tˆƒ ksÂt‚||ˆ d ƒˆd t| ˆ ƒ tˆƒ d ksút‚|ˆd ˆd d  ƒt	ˆƒ tˆƒ tˆƒ ks2t‚|ddˆd d   ƒt	ˆƒ tˆƒ tˆƒ ksjt‚|||d ˆd   ƒt	|ˆ ƒtˆƒ ksœt‚|ˆˆd |d   ƒt
|ˆ ƒtˆƒ ksÎt‚|||d |ˆ d   ƒt| ˆ ƒt	|ˆ ƒ tˆƒ kst‚||ˆ |d |ˆ d   ƒ|| t| ˆ ƒ t	|ˆ ƒ | t| ˆ ƒt
|ˆ ƒ   tˆƒ ks‚t‚|dˆd ˆd d   ƒˆtˆƒ t	ˆƒtˆƒ  ksÀt‚||ˆ |d ˆ| d  |d   ƒ|| | t	|ˆ | ƒ |t
|ˆ | ƒ   t| ˆ ƒ tˆƒ ||d   ksBt‚||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sÐt‚||ˆd |d   ƒ|t|ˆ ƒ tˆƒ | ks
t‚||ˆd |d   ƒ|t|ˆ ƒtˆƒ d|  t| ˆ ƒtˆƒ d|    ksft‚||ˆ ˆd |d   ƒ|t|ˆ ƒ tˆƒ ks t‚||ˆˆ|   ƒ|tˆƒ| t| ˆ ƒtˆƒ |   ksät‚|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	sVt‚|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	sÞt‚|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
sDt‚|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
sêt‚|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sbt‚||ˆ |tˆƒ |  ƒt|d ˆ |d  ƒ t|tˆƒ | ƒ d tˆƒ ks¾t‚||| 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\t‚|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s¾t‚||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s0t‚||| 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sÖt‚|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sTt‚|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sÖt‚|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sDt‚|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sÌt‚|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sÒt‚|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srt‚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,t‚|tˆƒ| ˆtˆƒ|   ƒdt|d ˆ ƒ t|tˆƒ ƒ d tˆƒ ks‚t‚|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t‚|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s¶t‚|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,t‚|dtˆ| ƒ ˆ|  ƒtˆƒdt|| ƒ t| ˆ ƒ tt|| ƒtˆƒ ƒ  ksŒt‚|dtˆd |d  ƒ ƒtˆƒtd|ˆ ƒ ksÄt‚|dtˆd |d  ƒ ƒtˆƒtd|ˆ ƒ ksüt‚|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s¸t‚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   rv   rw   rk   ñ  s    z+test_inverse_laplace_transform.<locals>.ILTc                    s   t | ˆ ˆddS )NTr   r|   r}   r~   rv   rw   ÚILTSô  s    z,test_inverse_laplace_transform.<locals>.ILTSc                    s   t t| ˆˆƒˆˆ iƒS r{   )r   r   r}   ©rE   rD   r?   ro   rv   rw   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   rg   r   r+   r,   r)   r(   r&   r%   r   r   r"   r1   r   r   r2   ZHalfr   r   r8   r7   )
ÚkÚnrl   rm   rn   Údrk   r   r   rq   rv   r€   rw   Útest_inverse_laplace_transformé  s$   ,2**"*20:8ÿ
<ÿ
Dÿ
F0 ÿÿ
ÿ$ÿ
@088822DHÿ
>&Rÿ
&:
ÿþ
þ:<ÿ
:(ÿ
&Bÿ
@ÿÿÿ
Bÿÿ"Jÿþ
þÿÿJÿÿÿ8ÿÿ*."ÿÿÿÿÿ:ÿÿ&:ÿÿÿ2&&ÿþþÿÿ ÿ
ÿÿþþÿÿ6ÿÿÿÿ*ÿÿÿÿ<"ÿÿÿÿ. ÿ*þþþ&ýüüÿÿÿ "ÿÿÿþ
ýÿÿÿ<:ÿÿÿþÿÿ.ÿÿ&2ÿþÿ,:ÿ*ÿÿÿÿ0ÿÿÿ 4ÿÿ ÿÿ ÿÿDÿÿþþýÿür†   c                  C   s¸  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s‚t‚|t| t ƒt t|ƒt	| | ƒks®t‚|t| t ƒ|t  t|ƒt| | |  ƒt	| | ƒ ksòt‚||t |d |t d   t|ƒt| | ƒt
|| ƒ t	|ƒ ks>t‚|t| t ƒt|  t|ƒ| | |d  t	| | ƒ t|ƒ ksŠt‚|t| t ƒttd d ƒ t|ƒt	| | ƒtd|| ƒ ksÒt‚|dtttd ƒ  t|ƒt	|ƒtt|ƒƒ kst‚|t| t ƒt|  t|ƒ|| |d  t	|| ƒ t|ƒ ksPt‚|t| t ƒtdtd  ƒ t|ƒt	|| ƒtd|| ƒ ks–t‚t||| tttd |d  ƒ |   ttd |d  ƒ t|ƒ t¡ƒt	|ƒt||| ƒ ksþt‚||| tttd |d  ƒ |   ttd |d  ƒ t|dd t¡t	|ƒt||| ƒ ksft‚|dtttd ƒ  t|ƒt	|ƒtt|ƒƒ ksœt‚|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sþt‚|td |
tƒ t|ƒtt|
tƒt|d ƒ|dfƒks8t‚|t
tƒt|ƒtt
tƒt|d ƒks`t‚|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s¸t‚t
tdƒd ƒ}||tdt ƒ t|ƒ|| |d ƒ ksöt‚|tdt ƒt|ƒttdt ƒt|d ƒks&t‚tddd}|t|t ƒt|ƒtt|t ƒt|d ƒksbt‚|td |d td   t|ƒ| t|| ƒ t	|ƒ | |ƒ ksªt‚|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t‚|t|t ƒt|ddtt|t ƒt|d ƒdfksRt‚tdddtddddfD ]F}	|t|	t ƒt|dd}
t|
tƒrœ|
d n|
}
|
j| kslt‚qld S )Nr   )r+   za b c dTr@   zn, rry   zt zrD   rE   c                 S   s   t t| ƒƒ t¡S r{   )r   r
   Úrewriter&   )Úexprrv   rv   rw   Úsimp_hypª  s    z4test_inverse_laplace_transform_old.<locals>.simp_hyprI   rQ   r   r\   rJ   rS   iùÿÿÿrU   rZ   ÚrFrK   Úz)Zextended_real)Z	imaginaryÚzero)Ú'sympy.functions.special.delta_functionsr+   r   r   r	   r?   r   rg   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   r…   r„   rŠ   ro   r‹   rD   rE   r‰   rr   rv   rv   rw   Ú"test_inverse_laplace_transform_old   s¤    ,D" ÿ
&ÿ
$ÿ
6"ÿ
$ÿ
< ÿÿþÿ:  ÿÿþÿ6*.ÿ
ÿ
(&ÿÿ
.00 ÿ
* ÿÿ
ÿ

ÿr’   c                  C   sÈ  t dƒ} t dƒ}t ddd}tt| ƒ| tƒtdtd  ƒ d t ddfksPt‚tt|| ƒ| tdd	tttt	t
 ƒ d|ƒdt|ƒtjkfks’t‚ttd| ƒ| tdd	ttd ƒt ddfksÂt‚ttd| ƒ| tdd	tttd ƒ td  ddfksút‚ttdtd  ƒ d t t|ƒ ¡ t|ƒt|ƒ ks6t‚tttd ƒt t| dd	 t¡t| ƒt| ƒ ksnt‚ttttd ƒ td  t| dd	 t¡ ¡ td| ƒt| ƒ  t¡ t¡ ¡ ksÄt‚d S )
Nrq   rl   ÚuT)ZpolarrQ   rI   r   r   )r   r   r4   r?   r   rg   r5   r.   r   r   r   r   r   rh   r   ri   r,   r‡   r6   r3   )rq   rl   r“   rv   rv   rw   Útest_expinté  sF    4&ÿÿÿ$ÿ
ÿÿþÿ  þþ ýÿr”   N)ZZsympy.integrals.laplacer   r   r   r   r   r   r   Zsympy.core.functionr	   r
   Z
sympy.corer   r   r   r   Zsympy.core.exprtoolsr   Zsympy.core.numbersr   r   r   Zsympy.core.relationalr   Zsympy.core.singletonr   Zsympy.core.symbolr   r   Zsympy.simplify.simplifyr   Z$sympy.functions.elementary.complexesr   r   Z&sympy.functions.elementary.exponentialr   r   r   Z%sympy.functions.elementary.hyperbolicr   r   r    r!   Z(sympy.functions.elementary.miscellaneousr"   Z$sympy.functions.elementary.piecewiser#   Z(sympy.functions.elementary.trigonometricr$   r%   r&   Zsympy.logic.boolalgr'   Z'sympy.functions.special.gamma_functionsr(   r)   r*   r   r+   r,   Z-sympy.functions.special.singularity_functionsr-   Z&sympy.functions.special.zeta_functionsr.   Z'sympy.functions.special.error_functionsr/   r0   r1   r2   r3   r4   r5   r6   Zsympy.functions.special.besselr7   r8   r9   r:   Zsympy.testing.pytestr;   r<   Zsympy.matricesr=   r>   Z	sympy.abcr?   rx   r†   r’   r”   rv   rv   rv   rw   Ú<module>   sH   $(
   I
 7
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