kill(all)$

complex variable : essential singularities @ poles ,f(s) example , version 1
peter.vlasschaert@gmail.com,04/08/2019

rem : f(s),'see below'

load(lrats);

How to find :inverse laplace transform.

example : '1'
=========

p1:f(s) =1/s* (sinh((sqrt(s)*x))/sinh((sqrt(s))));

domain;

wxplot2d([sinh(x)], [x,-5,5])$

wxplot2d([sinh(sqrt(s))], [s,-1,5])$

taylor : around : 0, see figure above
===========================

p1a:f1=num(rhs(p1));

p2a:f1(s)=taylor(rhs(p1a),x,0,3);

p1b:f2=denom(rhs(p1))/s;

p2b:f2(s)=taylor(rhs(p1b),s,0,2);

p1c:f3=1/s;

p2c:f3(s)=rhs(p1c);

laurent expansion : around 0

p3:f(s) =rhs(p2c)*(rhs(p2a)/rhs(p2b));

corrected: s/36=0, because :only need terms  : 1/s .

p3a:lratsubst([s=0],coeff(rhs(p3),x,3))*x^3;

p3b:coeff(rhs(p3),x,1)*x;

only :pole @ s= 0: exp(st)*f(s)
=======================

p3c:expand(p3a+p3b);

p3d:taylor(exp(s*t),t,0,2);

exp(st)*f(s)  , removes terms :s,s^2,s^3...
==============================

p4:p3c*p3d;

p4a:expand(p4);

p4b:collectterms(p4a,s^2,s);

removes : s,s^2 , from 'p4b'
-------------------------------------

p5:args(p4b);

p5a:lratsubst([p5[1] =0,p5[2]=0],p5);

p5b:sum(p5a[i],i,1,length(p5a));

coef : 1/ s => 'p6'
=============

p5b1:s*args(p5b);

p6:sum(p5b1[i],i,1,length(p5b1));

value :exp(st)*f(s)
=============

p6a:expand(1/s*p6);

fact: conversion
hyperbolic function <=> trignometric function
==================================

show : sinh(i*θ) = i*sin(θ)
===================

p7a:sinh(%i*θ) = %i*sin(θ);

p7b:exponentialize(p7a);

check :

p7c:is(p7b);

show : cosh(i*θ) =cos(θ)
====================

p8a:cosh(%i*θ) = cos(θ);

check:

p8b:is(p8a);

rem :f(s) has singularity  p1b
----------------------------------------
find s : ?  for singularities @ zero sinh(sqrt(s))

1e)  fact from above
----------------------------

p9:'sinh(%i*θ) = %i*'sin(θ);

zero's from sine function
==================

wxplot2d([sin(x)], [x,-2*%pi,2*%pi])$

for : integer number π we have zero value for function
=========================================

p9a:rhs(p1b);

relation between : 'p9' and 'p9a'
========================

p9b:%i*θ =sqrt(s);

solve for θ
=========

p9c:solve(p9b,θ);

p9c1:p9c[1];

roots for sine function :θ = n*π (see figure above)
=======================

p9b:θ=n*π;

find s: ?
=======
use : 'p9c1 and 'p9b'

p10:lratsubst([p9b],p9c1);

p10a:solve(p10,sqrt(s));

p10b:p10a^2;

p10c:p10b[1];

def : s0  is pole order  m
===================

p11:m1=(1/(m-1)!*'diff((s-s0)^m*f*exp(s*t),s,m-1));

p11a:ev(subst([m=1,f=rhs(p1)],rhs(p11)),diff);

p11b:lratsubst([s0=rhs(p10c)],p11a);

rem: l'hopital : ?  (s+%pi^2*n^2)/sinh(sqrt(s))

declare(n,integer);

assume(n≥0);

denom: p11b (part)
-------------------

p12:denom(p11b);

p12a1:p12/s;

p12a2:limit (p12a1,s,-n^2*π^2);

num:p11b(part)
---------------

p13:num(p11b);

p13a1:part(p13,1);

p13a2:limit (p13a1,s,-n^2*π^2);

result : 0/0 need  l'hopital rule : (fnum())'/(fdenom())',' =differentation  with respect to s

p14:'diff(p13a1,s)/'diff(p12a1,s);

p14a:p14=ev(p14,nouns);

final general limit : s → - %pi^2*n^2 for 'p14a'

p15:p11b*(p12a1/p13a1)*rhs(p14a);

p15a:limit (p15,s,-n^2*π^2);

final result :f(t)
===========

pole:@ s=-n^2*π^2
===============

p16:f1(t) = sum(p15a,n,1,∞);

pole:@ s=0
=========
use :p6, 'see above'

p17:p6;

p17a:f2(t) =limit (p17,s,0);

result :f(t)

p18:f(t) =rhs(p17a) +rhs(p16);

rem : sinh(i*θ) = i*sin(θ)
          cosh(i*θ) = cos(θ)
see : 'above'