If f(z) is analytic everywhere in the complex plane, it is called
entire
. Examples • 1/z is analytic except at z = 0, so the function is singular at that point. The functions zn, n a nonnegative integer, and ez are entire functions.
How do you know if a function is analytic everywhere?
If f(z) is analytic everywhere throughout some neighborhood of a point z = a, say inside a circle C :
|z − a| = R
, except at the point z = a itself, then z = a is called an isolated singular point of f(z). f(z) cannot be bounded near an isolated singular point.
Which functions are always analytic?
The
trigonometric functions, logarithm, and the power functions
are analytic on any open set of their domain.
Is the function f z )= E Z analytic?
We say f(z) is complex differentiable or rather analytic if and only if the partial derivatives of u and v satisfies the below given Cauchy-Reimann Equations. So in order to show the given function is analytic we have to check whether the function satisfies the above given Cauchy-Reimann Equations. …
e(iy)=ex(cosy+isiny)
Is FZ Z 2 analytic?
We see that f (z) = z
2
satisfies the Cauchy-Riemann conditions throughout the complex plane. Since the partial derivatives are clearly continuous, we conclude that f (z) = z
2
is analytic
, and is an entire function.
Which of the following is not analytic function?
C.R. equation is not satisfied. So,
f(z)=|z|2
is not analytic.
Is a constant function analytic?
Constant functions are
analytic
.
Is analytic function single valued?
A single-valued function is function that, for each point in the domain,
has a unique value in the range
. It is therefore one-to-one or many-to-one. independent of the path along which it is reached by analytic continuation (Knopp 1996).
Are all analytic functions holomorphic?
Every holomorphic function is analytic
. … In fact, f coincides with its Taylor series at a in any disk centred at that point and lying within the domain of the function. From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space.
Is f z )= sin z analytic?
Solution: When z = x + iy, sinz = sin xcosh y + icosxsinh y. … So
sin z is not analytic anywhere
. Similarly cos z = cosxcosh y + isinxsinhy = u + iv, and the Cauchy-Riemann equations hold when z = nπ for n ∈ Z. Thus cosz is not analytic anywhere, for the same reason as above.
How can you tell if a function is analytic?
A function
f (z) = u(x, y) + iv(x, y)
is analytic if and only if v is the harmonic conjugate of u.
Is EZ whole?
ez is not injective unlike real exponential. Since ez = ex cos y + iex sin y satisfies C-R equation on C and has continuous first order partial derivatives. Therefore
ez is an entire function
.
Is log z analytic?
Answer: The function Log
(z) is analytic except
when z is a negative real number or 0.
Why is z 2 not analytic?
(a) z = x + iy, |z|2 = x2 + y2, u = x2,
v = y2 ux = 2x = vy = 2y
Hence not analytic. The partial derivatives are continuous and hence the function is ana- lytic.
Is E 1 z analytic?
In this deleted neighborhood,
e1/z is analytic
. So for any point in this neighbourhood, we can expand ez first and then substitute 1/z in. As you can see in the Laurent expansion you gave, you can plug in any z arbitrarily close to zero, calculate the infinite sum, and get a finite and well defined value.
