Struggling with Complex Number Function? Need Help?

In summary: You are correct. Substitute the hint DaalChawal gave into the function definition of $f(z)$ and simplify it.Then substitute the limit value $z=\frac 12 i$ for $f(z)$ with $z\ne \frac 12 i$ and evaluate it.How far do you get?
  • #1
jaychay
58
0
lim 1.png
lim 2.png


Can you help me with this two questions
I am really struggle on how to do it
Please help me

Thank you in advance
 
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  • #2
Factorize $(4z^2 + 1) = (4z^2 - i^2)= (2z+i)(2z-i)$
 
  • #3
DaalChawal said:
Factorize $(4z^2 + 1) = (4z^2 - i^2)= (2z+i)(2z-i)$
How can I solve the limit
 
  • #4
jaychay said:
How can I solve the limit
Substitute the hint DaalChawal gave into the function definition of $f(z)$ and simplify it.
Then substitute the limit point $z=\frac 12 i$ for $f(z)$ with $z\ne \frac 12 i$ and evaluate it.
How far do you get?
 
Last edited:
  • #5
Klaas van Aarsen said:
Substitute the hint DaalChawal gave into the function definition of $f(z)$ and simplify it.
Then substitute the limit value $z=\frac 12 i$ for $f(z)$ with $z\ne \frac 12 i$ and evaluate it.
How far do you get?
On question 2
Can you tell me on how to find (a,b) that can make it continue at z=1/2i
 
  • #6
jaychay said:
On question 2
Can you tell me on how to find (a,b) that can make it continue at z=1/2i
Choose $a$ and $b$ such that $a+bi$ is the same as the limit value from question 1.
Did you find the limit value or are you stuck somewhere?
 
  • #7
Klaas van Aarsen said:
Choose $a$ and $b$ such that $a+bi$ is the same as the limit value from question 1.
Did you find the limit value or are you stuck somewhere?
[/QUOT
Did I do it correct ?
 

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  • #8
You made a mistake there bro
$2(\frac{i}{2})+i = 2i$
 
  • #9
jaychay said:
Did I do it correct ?
The first substitution is correct, but the parentheses are placed the wrong way in the substitution of $z=\frac 12 i$.
It should be:
$$\lim_{z\to\frac 12i} f(z) = \lim_{z\to\frac 12i} \frac{4z^2+1}{2z-i} = \ldots = \lim_{z\to\frac 12i} 2z+i = 2\left(\frac 12 i\right) +i=2i$$
 
  • #10
Tha
DaalChawal said:
You made a mistake there bro
$2(\frac{i}{2})+i = 2i$
Thank you bro
 
  • #11
Klaas van Aarsen said:
Substitute the hint DaalChawal gave into the function definition of $f(z)$ and simplify it.
Then substitute the limit point $z=\frac 12 i$ for $f(z)$ with $z\ne \frac 12 i$ and evaluate it.
How far do you get?
Can you please check me on my another post on complex graph equation problem on question 2 on how to find ( a, b )
 
  • #12
Klaas van Aarsen said:
The first substitution is correct, but the parentheses are placed the wrong way in the substitution of $z=\frac 12 i$.
It should be:
$$\lim_{z\to\frac 12i} f(z) = \lim_{z\to\frac 12i} \frac{4z^2+1}{2z-i} = \ldots = \lim_{z\to\frac 12i} 2z+i = 2\left(\frac 12 i\right) +i=2i$$
On question 2
Is ( a,b ) equal to ( 0,2 ) right ?
 
  • #13
Klaas van Aarsen said:
The first substitution is correct, but the parentheses are placed the wrong way in the substitution of $z=\frac 12 i$.
It should be:
$$\lim_{z\to\frac 12i} f(z) = \lim_{z\to\frac 12i} \frac{4z^2+1}{2z-i} = \ldots = \lim_{z\to\frac 12i} 2z+i = 2\left(\frac 12 i\right) +i=2i$$
Klaas van Aarsen said:
Substitute the hint DaalChawal gave into the function definition of $f(z)$ and simplify it.
Then substitute the limit point $z=\frac 12 i$ for $f(z)$ with $z\ne \frac 12 i$ and evaluate it.
How far do you get?
On question 2
Is ( a,b ) equal to ( 0,2 ) right ?
 

FAQ: Struggling with Complex Number Function? Need Help?

1. What are complex numbers?

Complex numbers are numbers that are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, equal to the square root of -1. They are used to represent quantities that have both a real and imaginary component.

2. How do I add or subtract complex numbers?

To add or subtract complex numbers, simply add or subtract the real and imaginary components separately. For example, (3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i.

3. What is the conjugate of a complex number?

The conjugate of a complex number is the number with the same real component, but with the imaginary component multiplied by -1. For example, the conjugate of 3 + 4i is 3 - 4i.

4. How do I multiply or divide complex numbers?

To multiply complex numbers, use the distributive property and the fact that i^2 = -1. For example, (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

5. How can I graph complex numbers?

Complex numbers can be graphed on a two-dimensional plane called the complex plane. The real component is plotted on the x-axis and the imaginary component is plotted on the y-axis. The point where the two axes intersect represents the number 0 + 0i, or simply 0. The distance from the origin to the point is the magnitude of the complex number.

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