# Estimate the magnitude of a line integral exp(iz) over a semicircle

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• hotvette
In summary, on page 158 in Fisher, it is stated that in applications of the Residue Theorem, there is a need to estimate the line integral of e^{iz} over a semicircle. The magnitude of e^{iz} is equal to e^{-R \sin(\theta)}, which can be proven using the modulus of a complex number.
hotvette
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Clarify statement in Complex Variables (S. Fisher) about magnitude of a line integral exp(iz) over a semicircle
Not homework, just trying to understand a statement in the book. On page 158 in Fisher, the following statement is made:
In these applications of the Residue Theorem, we often need to estimate the magnitude of the line integral of $e^{iz}$ over the semicircle $= Re^{i\theta}, \; 0 \le \theta \le \pi$. We note that:
$$\left\vert e^{iz} \right\vert = e^{\text{Re}(iz)} = e^{-R \sin(\theta)}$$
To understand the above, I did the following.
\begin{align}
&z = Re^{i\theta} = R \cos(\theta) + i R \sin(\theta)
\\
&iz = i R \cos(\theta) - R \sin(\theta)
\\
&e^{iz} = e^{i R \cos(\theta) - R \sin(\theta)} = e^{i R \cos(\theta)} e^{-R \sin(\theta)}
\\
&\left\vert e^{iz} \right\vert = \left\vert e^{i R \cos(\theta)} e^{-R \sin(\theta)} \right\vert
= \left\vert e^{i R \cos(\theta)} \right\vert \; \left\vert e^{-R \sin(\theta)} \right\vert
\\
&e^{i R \cos(\theta)} = e^{ix} && \text{where $x = R \cos(\theta)$, real}
\\
\therefore \, &\left\vert e^{i R \cos(\theta)} \right\vert = 1
&& \cos^2 x + \sin^2 x= 1
\\
&\left\vert e^{-R \sin(\theta)} \right\vert = e^{-R \sin(\theta)}
&& \text{since } e^{-R \sin(\theta)} \ge 0
\\
\therefore \, &\left\vert e^{iz} \right\vert = e^{-R \sin(\theta)} =e^{\text{Re}(iz)}
\end{align}
Is it correct?

Last edited:
Delta2
Yes (IMHO)

Thanks!

How does one determine a modulus of a complex number ?

Use that to prove ##|\bf ab | = |a|\,|b|##

I offer the following observation:$$\left| e^{iz} \right|= \sqrt{e^{iz}e^{-iz^*}}=\sqrt {e^{i(z-z^*)}}\\ =e^{\frac{iR}{2}((\cos(\theta)+i\sin(\theta))-(\cos(\theta)-i\sin(\theta)))}=e^{-R\sin(\theta)}$$

Got it !

## 1. What is a line integral?

A line integral is a mathematical concept that measures the integral of a function along a curve or path. It takes into account both the values of the function and the length of the curve.

## 2. What is the magnitude of a line integral?

The magnitude of a line integral is the absolute value of the line integral. It represents the overall size or magnitude of the integral, regardless of its direction.

## 3. What is a semicircle?

A semicircle is a half of a circle, formed by cutting a circle along its diameter. It has a curved edge and a straight edge, with a total of 180 degrees.

## 4. How do you estimate the magnitude of a line integral over a semicircle?

To estimate the magnitude of a line integral over a semicircle, you can break the curve into smaller segments and approximate the integral using numerical methods such as the trapezoidal rule or Simpson's rule. Alternatively, you can use a computer program to calculate the integral accurately.

## 5. What is the significance of exp(iz) in the line integral over a semicircle?

The function exp(iz) is a complex exponential function, where z is a complex number. It is often used in physics and engineering to represent oscillatory behavior. In the context of a line integral over a semicircle, it represents the function being integrated along the curve.

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