Question

For a complex variable z = x + iy, which of the following statements is true?
1. Both sin h Z and cos h Z are entire functions
2. Neither sin h Z nor cos h Z are entire functions.
3. sin h Z is entire but cos h Z is not an entire function
4. sin h Z is not entire but cos h Z is an entire function.

Answer 1

Correct Answer - Option 1 : Both sin h Z and cos h Z are entire functions

Explanation:

Analytic Function

If a complex function f(Z) is differentiable at point Z₀ and also differentiable at every point in some neighbourhood of a point Z₀ then the function f(Z) is called an analytic function at point Z = Z₀

Entire Function

If a function f(Z) is analytic at every point in the whole finite complex plane then the function f(Z) is called an entire function. As sin h Z and cos h Z are analytic everywhere in the complex plane so they are called entire functions.

 Analytic Function

• A function f(z) which is single-valued and possesses a unique derivative with respect to z at all points of the region R, is called the analytic function of z in that function.
• An analytic function is also called a regular function or a holomorphic function.
• A function that is analytic everywhere in the complex plane is known as an entire function.
• As a derivative of a polynomial exists at every point, a polynomial of any degree is an entire function.
• A point at which an analytic function ceases to possess a derivative is called a singular point of the function.