exponent turns field into zero amplitude

[yefj]

Hello,bellow we can see the field lines and its formula's.
for x=a/2 and z=lambda/4 we suppose to get Ex=0 Ez=Max
from E_x the cosine is zero
but E_z is zero also because the exponent will give us complex "j"
in both Ex and Ez there are exponent expressions with z variable.
How mathematickly this exponent can cause the whole expression to be zero?(as shown in the field lines bellow)
Thanks.

 

[wwfeldman]

Ex and Ez depend on sin and cos functions
for Ex, cos = 0 when pi/a *x = pi/2, 3pi/2, 5pi/2, etc
for Ez, sin = 0 when pi/a * z = 0, pi, 2pi, 3pi, etc

at x=a/2, pi/a * x = pi/a * a/2 = pi/2 so cos =0
at x = a, pi/a * x = pi/a * a = pi, so sin =0

as for "at z=lambda/4 we suppose to get Ex=0 Ez=Max ",
look at euler's formula

e^i x = cos ⁡ x + i sin ⁡ x

Euler's formula - Wikipedia

en.wikipedia.org

 

[yefj]

exponent can be zero only at minus infinity.
our complex exponent has absolute vale of 1 at every value
how exactly its zero?

 

[wwfeldman]

the zeros you are seeing in your field diagram in first post
are due to sin and cos, not due to the exponent

euler's formula e^i x = cos ⁡ x + i sin ⁡ x
has a real part, cos, and an imaginary part, sin

if x = - infinity, e^(-j*infinity) is NOT the same as e^(-infinity)

e^(-j*infinity) = cos ⁡ (-infinity) + i sin ⁡(-infinity) is not necessarily zero
whereas
e^(-infinity) = 0 is necessarily zero

 

[yefj]

yes so how exactly a complex exponent can be zero
there is no suc value