A "forgotten" theorem about complex numbers

[hgjf2]

If giving a function f:C->C where C is complex numbers set, and f(z)=f(z+k+l*i) where k,l<-R (real numbers) and z=a+bi.
Exist least on complex number z0 for f(z0) = 1/0 (infinity)

[Roquen]

I'm not understanding what you're saying.  You seem to be defining translation.   F(z) = z+t with z,t elements of C.

[hgjf2]

I'm not understanding what you're saying.  You seem to be defining translation.   F(z) = z+t with z,t elements of C.

f(z) is defined here as periodical function with two complex vectors orthogonals. Than f(z)=f(z+km+lni) like sin(z)=sin(z+2k[pi]) and exp(z)=exp(z+2ki[pi]).
k,l<-Z (integers) and m,n<-R (real parameters) and i=sqrt(-1).
More explains:
A function like f(z) from my definition is integral(sqrt(sin(z))) as example.
f(z)=f(z+m)=f(z+2*m)=... =f(z+ni)=f(z+m+ni)=f(z+2*m+ni)=... =f(z+2*ni)=f(z+m+2*ni)=f(z+2*m+2*ni)=...

[kram1032]

So you're asking wether there always exists a complex number z0 such that f(z0) has a pole of 1-st grade at z0, given the function f is periodic in both the real and the imaginary direction?