peter.vlasschaert@gmail.com,20/09/2017
=> infinitely  many singularities. (from f(z)) ,version 1.0

relation between : {sinh(z),cosh(z) } and { sin(z),cos(z)}
(1)    cosh(z) = cos(i*z)
(2)   i*sinh(z) = sin(i*z)

Why the above :relation (1),(2)

kill(all)$

p1:cosh(z)      =  cos(%i*z);
p2:%i*sinh(z)  =  sin(%i*z);

p3: ( lhs(p1)) ^2= (rhs(p1))^2;
p4: ( lhs(p2)) ^2= (rhs(p2))^2;
p5 :p3+p4;

trigsimp((p5));

(cosh(z))^2-(sinhz(z))^2 = 1
cos(i*z)^2+sin(i*z)^2       = 1

load(to_poly_solve)$

all zeros :sin(x)   

p6: f(x):=sin(x);

p7:to_poly_solve([f(x)], [x])$
p8:part(p7,1,1,2,1);
p9:part(p7,1,1,2,2);
p10:part(subst(k1, p9, p7),1,1,2);

sin(x) = 0   =>    x=pi*k1 , k1 = ...,-2,-1,0,1,2,.......

all zeros:cos(x)

p11:sin(x+%pi/2);

p12:g(x):=sin(x+%pi/2);

p13:to_poly_solve([g(x)], [x])$
p14:part(p13,1,1,2);
p15:num(p14);
p16:denom(p14);
p17:subst(k2, %z30, p15)/p16;

cos(x) = 0  =>     x=(2*pi*k2+pi)/2 ,k2=....-2,-1,0,1,2,...

check : above solution => cos(x) = 0

p18:cos(p17);
p19:ev(cos(p17),k2=-3);

find all singularities f(s) = (cosh(x*sqr(s))/(s*cosh(sqr(s))
sqr =()^(1/2)

 q1:u=x*(s)^(1/2);
 q2: v=(s)^(1/2);
p21:cosh(cc);
p22:taylor(p21, cc, 0, 6);
p23:subst(part(q1,2), cc, p22);
p24:subst(part(q2,2), cc, p22);

p25:fss= p23/(s*p24);

simple pole 's=0'
infinite many poles ' cosh(s^(1/2)) '

find : infinite many poles ' cosh(s^(1/2)) '

p26:ev(p21,cc=part(q2,2))=0;
p27:exponentialize(p26);
p28:factor(p27);
p29:num(part(p28,1))=rhs(p28);
p30:part(p29,1,2)-1=-1;
p31:solve(p29,s);
p32:solve(p28,s);

rem :e^(i*pi) = -1 = e^(i(pi+2*kk*pi))

p33:2*s^(1/2)=%i*(%pi+2*kk*%pi);
p34:factor(part(solve(p33,s),1));

check : cosh(s^(1/2))=0 ,kk=-2

ev(ratsubst(rhs(p34), s, cosh(s^(1/2))),kk=-2);

=>  cosh(z) = cos(i*z)
find : cosh(s^(1/2)) 
rem : result ' p17 ' => used

p35:s^(1/2) =%i*p17;
p36:factor((p35)^2);

check : cosh(s^(1/2))=0 ,k2=-2

ev(ratsubst(rhs(p36), s, cosh(s^(1/2))),k2=-2);

rem : about sinh(z)

analog :    result => 
sinh(s^(1/2))    = 0
i*sinh(s^(1/2))  = 0
sin(i*s^(1/2))     = 0

find : sinh(s^(1/2)) = 0

p37:s^(1/2) =%i*p10;
p38:(p37)^2;

check : sinh(s^(1/2)) = 0

ev(ratsubst(rhs(p38), s, sinh(s^(1/2))),k1=-2);