kill(all)$

find zero of cosh(z)= 0 ,find z
peter.vlasschaert@gmail.com,21/09/2017

find :why cosh(s^1/2) = 0  ,find other solution

f(z):=cosh(z);

p1:f(s^(1/2))=0;

method 1 :

p2:exponentialize(f(sqrt(s))=0);
p3:solve(p2,s);
p4:p3[1]^(1/2);
p5:p3[2]^(1/2);
p6:lhs(p4) = radcan(rhs(p4));
p7:lhs(p5) = radcan(rhs(p5));

check :  cosh(sqrt(s))=0

why no result : because branch cut complex plane  log(z ) we get
                               no direct evaluation. 
reason : use branch cut because otherwise more than 'one' value.

p8:f(rhs(p6));
p9:f(rhs(p7));

How to solve in maxima, convert to polar form for log(z)
menu => simplify => complex simplification => convert to polarform

1e)  log(-%i)   =>  convert polarform => ratsimp => f(z)

p10:polarform(log(-%i));
p11:ratsimp(p10);
p12:f(p11);

2e)  log(%i)   =>  convert polarform => ratsimp => f(z)

p13:polarform(log(%i));
p14:ratsimp(p13);
p15:f(p14);

rem :                                     +/- sqrt(s)   => +/-(-i*pi/2) and +/-(+pi/2)
=>   see also file :  infiniteleymanysingularitiesversion1.wxmx

1e)  full :solution log(-%i) => log(z) = log(abs(-i))-(i*pi)/2
        => cosh(log(z)) = 0
use : cosh(z) = cos(i*z) ,show the above.

p16:log(abs(-%i))+p11;
p17:f(p16);
p18: cos(%i*p16);

2e)  full :solution log(%i) => log(z) = log(abs(i))+(i*pi)/2
        => cosh(log(z)) = 0
use: cosh(z) = cos(i*z), show the above.

p19:log(abs(%i))+p14;
p20:f(p19);
p21: cos(%i*p19);

method 2 :

full definition :      log(z) = w0  + integrate(1/z,z,z0,z)  
                                  z0 = exp(w0)
info  :                       integrate(1/z,z,z0,z)  line integral from z0 to z
                                  Branch Cut ( example : 0 to -infinity)

find : log(i)  'with above definition'

1e)  need B = branch cut  ' 0  to   - infinity '
        =>     otherwise multiple values log(i)

2e) we need choose value w0

w0:0;

3e) calculation of z0

z0 : exp(w0);

4e) log(z) ?

'log(z) = 'w0+'integrate(1/ε,ε,z0,z);

5e)  positive x axis  to complex value i
        How to :use line integral along quarter of a circle 
                        from 1 to i .
                       parameterization : z = ρ*exp(i*theta)
                                                           unit circle :  ρ = 1
complex plane : x axis = real part
                               y axis = imaginary part
info :                     complex plane = gauss plane.

lower boundary integral

z0 = 1  => theta1  ?

p22: 1=exp(%i*theta1);
p23:solve(p22,theta1)[1];

upper boundary integral

z = i  => theta2    ? 

p24:%i = exp(%i*theta2);
p25:solve(p24,theta2)[1];
p26:lhs(p25)=polarform(rhs(p25));

 find : ε, dε    'parametrization'

ε:exp(%i*theta);
dε :diff(exp(%i*theta),theta,1);

find : log(i) ,log(-i) ?

p27:'log(i)   = w0+'integrate(dε/ε,θ,rhs(p23),rhs(p26));
p28:'log(-i) = w0+'integrate(-dε/ε,θ,rhs(p23),rhs(p26));

?   cosh(log(i)),cosh(log(-i))

p29:cosh(rhs(p27));
p30:cosh(rhs(p28));

rem : log(z) = ln(abs(z))+i*arg(z)
           log(z) = ln(abs(z)) + i*θ
other definitions