- #1
alvielwj
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How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C.
π is pi.
for all z belong to C.
π is pi.
How to show "if ..."alvielwj said:How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C.
π is pi.
An entire function is a complex-valued function that is defined and analytic (differentiable) at every point in the complex plane. This means that it has no singularities or poles in its domain.
To show boundedness of an entire function, we can use Liouville's theorem, which states that any entire function that is bounded on the entire complex plane must be a constant function. Therefore, if we can show that the given function is bounded, we can conclude that it is a constant function.
This means that the given function has period 2π, which means that it repeats itself every 2π units in the complex plane. This is similar to the concept of periodic functions in real analysis.
The periodicity of the function does not affect its boundedness. Even though the function repeats itself every 2π units, it still remains an entire function, and its boundedness can be determined using Liouville's theorem.
One example of an entire function with period 2π is f(z) = eiz, also known as the exponential function. This function repeats itself every 2π units in the complex plane, but it is still bounded, as it takes on values between 0 and 1.