For w=Ln z, u=loge|z| and v=Arg z . The semi-circle |z|=1, y>0 may also be described by r=1 0<θ<

step 1 Hello everyone here we have the circular integral over C for f z d z is equal to 0. Now this imp

For w=Ln z, u=loge|z| and v=Arg z . The semi-circle |z|=1,...

For $w=\operatorname{Ln} z, u=\log _{e}|z|$ and $v=\operatorname{Arg} z .$ The semi-circle $|z|=1, y>0$ may also be described by $r=1$ $0<\theta<\pi .$ Therefore $u=0$ and $0<v<\pi .$ The image is therefore the open line segment from $z=0$ to $z=\pi i.$

Key Concepts

Polar Representation of Complex Numbers

Expressing complex numbers in polar form, z = r·e^(i?), where r is the modulus and ? is the argument, simplifies the analysis of functions and curves in the complex plane. This representation is fundamental in understanding how functions like the logarithm transform the components of a complex number independently, with the modulus mapping through the real logarithm and the argument transforming into the imaginary component.

Mapping of Curves under Complex Functions

Mapping curves using functions like the complex logarithm involves analyzing how the modulus and argument of the original complex number are transformed into the real and imaginary parts of the new complex number. This concept is central in complex analysis, as it provides insight into how geometric figures, such as circles and arcs, are translated into their images under a given complex function.

Complex Logarithm

The complex logarithm is an extension of the real logarithm to complex numbers, defined as Ln z = log|z| + iArg z. This formula expresses the complex number in terms of its magnitude (modulus) and direction (argument). Due to the periodicity of the argument, the complex logarithm is inherently a multi-valued function, which often requires the specification of a particular branch to yield a unique value.

Branch Cuts in Complex Analysis

Due to the multi-valued nature of functions such as the complex logarithm, branch cuts are necessary to define a single continuous branch of the function. By restricting the range of the argument, typically to a principal value (like 0 to 2? or -? to ?), branch cuts ensure that the function is well-defined and continuous over the domain of interest, which is crucial for correctly mapping regions in the complex plane.

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