Math / Calculus

Questionif the taylor expansion of the function: $f(z)=\frac{1}{z^{2}+2 z+5}$ is written around $z_{0}=1+i$ , what's the radius of convergence?

Studdy Solution

STEP 1

1. The function $f(z) = \frac{1}{z^2 + 2z + 5}$ is a rational function.
2. The radius of convergence for a Taylor series expansion is determined by the distance from the center of expansion to the nearest singularity.
3. Singularities of a function occur where the denominator is zero.

STEP 2

1. Identify the singularities of the function.
2. Calculate the distance from the center of expansion $z_0 = 1 + i$ to the nearest singularity.
3. Determine the radius of convergence based on this distance.

STEP 3

Identify the singularities of $f(z) = \frac{1}{z^2 + 2z + 5}$ .
To find the singularities, solve the equation:
$z^2 + 2z + 5 = 0$
This is a quadratic equation, and we can use the quadratic formula:
$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 1$ , $b = 2$ , and $c = 5$ .

STEP 4

Calculate the discriminant $b^2 - 4ac$ :
$b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16$
Since the discriminant is negative, the roots are complex.

STEP 5

Calculate the roots using the quadratic formula:
$z = \frac{-2 \pm \sqrt{-16}}{2}$ $z = \frac{-2 \pm 4i}{2}$ $z = -1 \pm 2i$
The singularities are at $z = -1 + 2i$ and $z = -1 - 2i$ .

STEP 6

Calculate the distance from the center of expansion $z_0 = 1 + i$ to each singularity.
For $z = -1 + 2i$ :
$\text{Distance} = |(1 + i) - (-1 + 2i)| = |2 - i| = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$
For $z = -1 - 2i$ :
$\text{Distance} = |(1 + i) - (-1 - 2i)| = |2 + 3i| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$

STEP 7

The radius of convergence is the distance to the nearest singularity. Therefore, the radius of convergence is:
$\boxed{\sqrt{5}}$
The radius of convergence is $\boxed{\sqrt{5}}$ .

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Questionif the taylor expansion of the function: $f(z)=\frac{1}{z^{2}+2 z+5}$ is written around $z_{0}=1+i$ , what's the radius of convergence?

Studdy Solution

STEP 1

1. The function $f(z) = \frac{1}{z^2 + 2z + 5}$ is a rational function.
2. The radius of convergence for a Taylor series expansion is determined by the distance from the center of expansion to the nearest singularity.
3. Singularities of a function occur where the denominator is zero.

STEP 2

1. Identify the singularities of the function.
2. Calculate the distance from the center of expansion $z_0 = 1 + i$ to the nearest singularity.
3. Determine the radius of convergence based on this distance.

STEP 3

Identify the singularities of $f(z) = \frac{1}{z^2 + 2z + 5}$ .
To find the singularities, solve the equation:
$z^2 + 2z + 5 = 0$
This is a quadratic equation, and we can use the quadratic formula:
$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 1$ , $b = 2$ , and $c = 5$ .

STEP 4

Calculate the discriminant $b^2 - 4ac$ :
$b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16$
Since the discriminant is negative, the roots are complex.

STEP 5

Calculate the roots using the quadratic formula:
$z = \frac{-2 \pm \sqrt{-16}}{2}$ $z = \frac{-2 \pm 4i}{2}$ $z = -1 \pm 2i$
The singularities are at $z = -1 + 2i$ and $z = -1 - 2i$ .

STEP 6

STEP 7

The radius of convergence is the distance to the nearest singularity. Therefore, the radius of convergence is:
$\boxed{\sqrt{5}}$
The radius of convergence is $\boxed{\sqrt{5}}$ .

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Questionif the taylor expansion of the function: f(z)=1z2+2z+5f(z)=\frac{1}{z^{2}+2 z+5}f(z)=z2+2z+51​ is written around z0=1+iz_{0}=1+iz0​=1+i, what's the radius of convergence?

Studdy Solution

STEP 1

STEP 2

1. Identify the singularities of the function.2. Calculate the distance from the center of expansion z0=1+i z_0 = 1 + i z0​=1+i to the nearest singularity.3. Determine the radius of convergence based on this distance.

STEP 3

STEP 4

Calculate the discriminant b2−4ac b^2 - 4ac b2−4ac:b2−4ac=22−4⋅1⋅5=4−20=−16 b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16 b2−4ac=22−4⋅1⋅5=4−20=−16Since the discriminant is negative, the roots are complex.

STEP 5

Calculate the roots using the quadratic formula:z=−2±−162 z = \frac{-2 \pm \sqrt{-16}}{2} z=2−2±−16​​ z=−2±4i2 z = \frac{-2 \pm 4i}{2} z=2−2±4i​ z=−1±2i z = -1 \pm 2i z=−1±2iThe singularities are at z=−1+2i z = -1 + 2i z=−1+2i and z=−1−2i z = -1 - 2i z=−1−2i.

STEP 6

STEP 7

The radius of convergence is the distance to the nearest singularity. Therefore, the radius of convergence is:5 \boxed{\sqrt{5}} 5​​ The radius of convergence is 5 \boxed{\sqrt{5}} 5​​.

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Questionif the taylor expansion of the function: $f(z)=\frac{1}{z^{2}+2 z+5}$ is written around $z_{0}=1+i$ , what's the radius of convergence?

1. Identify the singularities of the function.
2. Calculate the distance from the center of expansion $z_0 = 1 + i$ to the nearest singularity.
3. Determine the radius of convergence based on this distance.

Calculate the discriminant $b^2 - 4ac$ :
$b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16$
Since the discriminant is negative, the roots are complex.

Calculate the roots using the quadratic formula:
$z = \frac{-2 \pm \sqrt{-16}}{2}$ $z = \frac{-2 \pm 4i}{2}$ $z = -1 \pm 2i$
The singularities are at $z = -1 + 2i$ and $z = -1 - 2i$ .

The radius of convergence is the distance to the nearest singularity. Therefore, the radius of convergence is:
$\boxed{\sqrt{5}}$
The radius of convergence is $\boxed{\sqrt{5}}$ .