# Is log z a holomorphic?

## Is log z a holomorphic?

In other words if f(z) is holomorphic, and we can define a continuous log f(z), then log f(z) is automatically holomorphic. dt = iθ. So the principal branch of the logarithm is given by log z = log r + iθ, where θ ∈ (−π, π).

How do you solve for log z?

Example 1: Calculate logz for z=−1−√3i. with n∈Z. The principal value of logz is the value obtained from equation (2) when n=0 and is denoted by Logz. Thus Logz=lnr+iΘ.

What is the derivative of log z?

Logz = lnr + θi, −π<θ<π, partial derivatives of its real and imaginary parts are ∂u ∂r = 1 r , ∂v ∂θ = 1, ∂u ∂θ = 0, ∂v ∂r = 0. Thus, Logz is analytic in the domain |z| > 0, −π < Argz<π. It is defined for all z = 0, but analytic only in the aforementioned domain.

### What is logarithm of complex number?

🔗 The logarithm function (for complex numbers) is an example of a multiple-valued function. All of the multiple-values of the logarithm have the same real part lnr ⁡ and the imaginary parts all differ by 2π. 2 π . 🔗

What is arg of z?

In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as. in Figure 1.

Why log z is analytic or not?

Answer: The function Log(z) is analytic except when z is a negative real number or 0.

#### What is arg z1 z2?

If z2 = 0, then arg(z1/z2) = arg(z1) − arg(z2). If z = a + bi, the conjugate of z is defined as z = a − bi, and we have the following properties: |z| = |z|, arg z = − arg z, z1 + z2 = z1 + z2, z1 − z2 = z1 − z2, z1z2 = z1z2, Re z = (z + z)/2, Im z = (z − z)/2i, zz = |z|2.

Is 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.

Is SINZ multivalued?

Yes, the complex arcsine function is, indeed, multivalued – but so too is, strictly speaking, its real counterpart. That sin−1(x), for x∈R, ranges in (−π,π] is an arbitrary convention.