What is the principal branch of log z?

What is the principal branch of log z?

NOTE The principal BRANCH of log z is defined to be equal to ln|z| + iθ where −π<θ<π. The principal VALUE of log z is defined to be equal to ln|z| + iθ where −π<θ ≤ π.

What is principal branch in complex analysis?

In mathematics, a principal branch is a function which selects one branch (“slice”) of a multi-valued function. Most often, this applies to functions defined on the complex plane.

What is complex logarithm function?

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number z, defined to be any complex number w for which ew = z. Such a number w is denoted by log z.

How do you find the principal branch?

The principal branch of arg(z) is between −π and π, so Arg(i)=π/2. Therefore, the value of log(i) from the principal branch is iπ/2.

How do you find the branch of a log?

Now define, f(z)=ln|z|+Arg(z)i z∈C. Function f is called a branch of the Logarithm, since f is analytic on C−[0,∞) (quite literally we can prove limz→z0f(z)−f(z0)z−z0 exists for every z0∈C−[0,∞)) and (more importantly), ef(z)=z ∀ z∈C−[0,∞).

What is principal branch value?

The principal branch of an analytic multivalued function, also called a principal sheet, is a single-valued “slice” (i.e., branch) of the function chosen that is for convenience in referring to a specific canonical value (a so-called principal value) of the function for each complex .

How do you find the principal value in a complex analysis?

The principal value is simply what we get when we adjust the argument, if necessary, to lie between -π and π. For example, z = 2e5 i/4 = 2e-3 i/4, subtracting 2π from the argument 5π/4, and the principal value of the argument of z is -3π/4.

How do you find the principal value of a function?

The principal value of the trigonometric function can be calculated by first breaking the angle into a sum of a larger angle and the acute angle. Further after finding the value of the acute angle for the trigonometric function, we can find the principal solution of the trigonometric function.

What do you mean by branch point and branch cut for a complex function?

Definition 1.1 The point Ю0 is called a branch point — for the complex (multiple) valued function. (Ю) — if the value of (Ю) does not return to its initial value as a closed curve around the point is. traced (starting from some arbitrary point on the curve), in such a way that varies continuously as the path is traced.

How do you calculate the principal value?

Principal Amount Formulas We can rearrange the interest formula, I = PRT to calculate the principal amount. The new, rearranged formula would be P = I / (RT), which is principal amount equals interest divided by interest rate times the amount of time.

What is the principal value of a function?

In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive real number.

How do you find the principal value of a branch?

By definition, the range of the principal value branch of y = sec-1 x is [0,π ]\ {π /2} . Let us find y in [0,π ] – {π/2} such that sec y = -2 . But, sec y = −2 ⇒ cos y = − 1/2 . Now, cos y =- 1/2 = -cos π/3 = cos (π – π/3 ) = cos 2π/3 .

What is the principal part of a function?

In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.