- #1

jaychay

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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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- MHB
- Thread starter jaychay
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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

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

DaalChawal

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Factorize $(4z^2 + 1) = (4z^2 - i^2)= (2z+i)(2z-i)$

- #3

jaychay

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How can I solve the limitDaalChawal said:Factorize $(4z^2 + 1) = (4z^2 - i^2)= (2z+i)(2z-i)$

- #4

I like Serena

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MHB

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Substitute the hint DaalChawal gave into the function definition of $f(z)$ and simplify it.jaychay said:How can I solve the limit

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

jaychay

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- 0

On question 2Klaas 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?

Can you tell me on how to find (a,b) that can make it continue at z=1/2i

- #6

I like Serena

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MHB

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Choose $a$ and $b$ such that $a+bi$ is the same as the limit value from question 1.jaychay said:On question 2

Can you tell me on how to find (a,b) that can make it continue at z=1/2i

Did you find the limit value or are you stuck somewhere?

- #7

- #8

DaalChawal

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You made a mistake there bro

$2(\frac{i}{2})+i = 2i$

$2(\frac{i}{2})+i = 2i$

- #9

I like Serena

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MHB

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The first substitution is correct, but the parentheses are placed the wrong way in the substitution of $z=\frac 12 i$.jaychay said:Did I do it correct ?

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

jaychay

- 58

- 0

Tha

Thank you broDaalChawal said:You made a mistake there bro

$2(\frac{i}{2})+i = 2i$

- #11

jaychay

- 58

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Can you please check me on my another post on complex graph equation problem on question 2 on how to find ( a, b )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?

- #12

jaychay

- 58

- 0

On question 2Klaas 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:

$$\lim_{z\to\frac 12i} f(z) = \lim_{z\to\frac 12i} \frac{4z^2+1}{2z-i} = \ldots = \lim_{z\to\frac 12i} 2z+i = 2\left(\frac 12 i\right) +i=2i$$

Is ( a,b ) equal to ( 0,2 ) right ?

- #13

jaychay

- 58

- 0

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 2Klaas 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?

Is ( a,b ) equal to ( 0,2 ) right ?

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.

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.

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.

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.

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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