Limits of z Approaching i Along Unit Circle |z|=1

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In summary, we are considering the limit limz→i(|z| + iArg(iz)) and looking at what value it approaches as z approaches i along the unit circle |z| = 1 in the first and second quadrants. When z is in the first quadrant, iz is in the second quadrant due to the rotation of 90 degrees when multiplied by i. In the first quadrant, the limit approaches 1 + i∏, while in the second quadrant, it approaches 1 − i∏.
  • #1
nickolas2730
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1. Consider the limit limz→i(|z| + iArg(iz))


2. (i) What value does the limit approach as z approaches i along the
unit circle |z| = 1 in the first quadrant?
(ii) What value does the limit approach as z approaches i along the
unit circle |z| = 1 in the second quadrant?



3. solutions are as follow:
(i) Note that iz = −y + ix. When z is in the first quadrant, iz is in the second quadrant. Thus, we have
limz→i(|z| + iArg(iz)) = 1 + i∏
when z approaches i along the unit circle |z| = 1 in the first quadrant.
(ii)Similarly, when z approaches i along the unit circle |z| = 1 in the
second quadrant,
limz→i(|z| + iArg(iz)) = 1 − i∏

My question is (i) why is it in second quadrant when z is in first quadrant?
is it because the real part is now -y and imaginary part is x which -y is the x-axis and x is the y-axis?
(ii)Why Arg(iz)=∏, when it is in second quadrant?

Thanks!
 
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  • #2
nickolas2730 said:
1. Consider the limit limz→i(|z| + iArg(iz))


2. (i) What value does the limit approach as z approaches i along the
unit circle |z| = 1 in the first quadrant?
(ii) What value does the limit approach as z approaches i along the
unit circle |z| = 1 in the second quadrant?

3. solutions are as follow:
(i) Note that iz = −y + ix. When z is in the first quadrant, iz is in the second quadrant. Thus, we have
limz→i(|z| + iArg(iz)) = 1 + i∏
when z approaches i along the unit circle |z| = 1 in the first quadrant.
(ii)Similarly, when z approaches i along the unit circle |z| = 1 in the
second quadrant,
limz→i(|z| + iArg(iz)) = 1 − i∏

My question is (i) why is it in second quadrant when z is in first quadrant?
is it because the real part is now -y and imaginary part is x which -y is the x-axis and x is the y-axis?
(ii)Why Arg(iz)=∏, when it is in second quadrant?
Any z on the unit circle can be represented as [itex]z= e^{i\theta}[/itex] so [itex]iz= e^{i\pi/2}e^{i\theta}= e^{i(\theta+ \pi/2}[/itex]. Multiplying by i "rotates" the number 90 degrees. When z is in the first quadrant, iz is in the second quadrant. When z is in the second quadrant iz is in the third quadrant.



Thanks!
 

FAQ: Limits of z Approaching i Along Unit Circle |z|=1

What is the significance of "Limits of z Approaching i Along Unit Circle |z|=1"?

The term "Limits of z Approaching i Along Unit Circle |z|=1" refers to the behavior of a complex number z as it approaches the imaginary number i on the unit circle, where the absolute value of z is equal to 1. This is important in understanding the behavior of complex functions and their limits.

What is the unit circle?

The unit circle is a circle on the complex plane with a radius of 1, centered at the origin. It is used to represent the complex numbers with magnitude and direction, where the direction is determined by the angle formed between the positive real axis and the complex number.

Why is the unit circle important in complex analysis?

The unit circle is important in complex analysis because it allows for the representation of complex numbers in terms of their magnitude and direction, which is crucial in understanding their behavior and limits. It also simplifies calculations and equations involving complex numbers.

What is the limit of z approaching i along the unit circle |z|=1?

The limit of z approaching i along the unit circle |z|=1 is equal to i. This means that as z gets closer and closer to i on the unit circle, the value of z will approach the value of i. This can also be represented as lim z→i z = i.

What is the geometric interpretation of the limit of z approaching i along the unit circle |z|=1?

The geometric interpretation of the limit of z approaching i along the unit circle |z|=1 is that as z gets closer to i on the unit circle, the angle formed between the positive real axis and z approaches 90 degrees. This is because i is located on the positive imaginary axis, which is perpendicular to the real axis. Therefore, the limit represents the direction in which z approaches i on the unit circle.

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