Question

Evaluate (in general) and plot in the complex plane, indicating which is the principal value: (a) ( facia ) (b) Gin peret (c) lif(factor) (f) i (¹/³)+ᶦ (g) (1+i) (¹/⁴)-ᶦ (h) tan(1/2i in 1+I/1-i

Answer 1

(a)
`\( e^(i\pi) = -1 \)`

(b)
`\( \text{Gin peret} = e^(i\pi/2) = i \)`

(c)
`\( \text{lif(factor)} = e^(i\pi/2) = i \)`

(f)
`\( i^(1/3) + i = \sqrt[3]{-1} + i \)`

(g)
`\( (1 + i)^(1/4) - i = e^(i\pi/8) - i \)`

(h)
`\( \tan\left((1)/(2)i\right) = -i \)`

Explanation:

In complex analysis, the principal values of expressions involving complex numbers are often determined by considering the properties of exponential functions and trigonometric functions. Let's evaluate each expression:

(a)
`\( e^(i\pi) = -1 \)`:

This result is derived from Euler's formula,
`\( e^(ix) = \cos(x) + i\sin(x) \)`, where
`\( x = \pi \) yields \( e^(i\pi) = -1 \)`.

(b)
`\( \text{Gin peret} = e^(i\pi/2) = i \)`:

Again, using Euler's formula,
`\( e^(i\pi/2) \)` corresponds to rotating a point in the complex plane by
`\(\pi/2\)` radians, resulting in the imaginary unit ( i ).

(c)
`\( \text{lif(factor)} = e^(i\pi/2) = i \)`:

Similar to the previous case,
`\( \text{lif(factor)} \)` is
`\( e^(i\pi/2) \)`, equating to ( i ).

(f)
`\( i^(1/3) + i = \sqrt[3]{-1} + i \)`:

Taking the cube root of ( i ) and adding ( i ) yields the desired result.

**(g)
`\( (1 + i)^(1/4) - i = e^(i\pi/8) - i \)`:

Applying the fourth root to ( 1 + i ) and subtracting ( i ) results in
`\( e^(i\pi/8) - i \)`.

(h)
`\( \tan\left((1)/(2)i\right) = -i \)`:

Evaluating the tangent of
`\( (1)/(2)i \)` leads to the principal value of (-i).

Answer 2

To evaluate and plot complex numbers on the complex plane, we need to represent each number as a combination of its real and imaginary parts. For the given expressions, we can determine their general form but not the exact values without specific information. However, we can still represent them in the complex plane. Examples for (a), (b), (c), (f), (g), and (h) are provided.

Step-by-step explanation:

To evaluate and plot the given complex numbers in the complex plane, we need to represent each number as a combination of its real and imaginary parts.

(a) To evaluate (facia), we need to find the value of facia.

Without the specific values for a and c, we cannot determine the exact value of this expression.

However, we can still represent it in the complex plane as (a + ci), where a is the real part and ci is the imaginary part.

(b) To evaluate Gin peret, we need to find the value of Gin peret.

Again, without the specific values for Gin and peret, we cannot determine the exact value.

However, we can represent it as (Gin + peret i) in the complex plane.

(c) To evaluate lif(factor), we need the specific values for lif and factor.

Without those values, we cannot determine the exact value.

Nevertheless, we can still represent it in the complex plane as (lif + factor i).

(f) To evaluate i^(1/3) + i, we can simplify it by using the rules of exponents to find the principal value.

Since i raised to the power of 4 equals 1, we can rewrite i as i^4.

Taking the cube root of both sides, we get i = (i^4)^(1/3), which simplifies to i = i^(4/3).

Plugging this value into the original expression, we have i^(1/3) + i = (i^(4/3))^(1/3) + i = (i^(4/9)) + i.

(g) To evaluate (1+i)^(1/4) - i, we can follow a similar process.

First, we rewrite 1+i as (1+i)^4 = (1+i)^2 * (1+i)^2.

Expanding this expression, we have (1+i)^4 = (1+2i+i^2) * (1+2i+i^2).

Simplifying further, we get (1+i)^(1/4) - i = [(1+2i+i^2)^2]^(1/4) - i = [(1+4i^2+4i)^2]^(1/4) - i = [(1+4(-1)+4i)^2]^(1/4) - i = (4i)^2^(1/4) - i = 4^(1/4) * i - i = 4^(1/4) * i - 1 * i = (4^(1/4) - 1) * i.

(h) To evaluate tan(1/2i) in 1+i/1-i, we start by using the identity tan(z) = sin(z)/cos(z).

Plugging in the value of z as 1/2i, we have tan(1/2i) = sin(1/2i)/cos(1/2i).

Using the Euler's formula e^(ix) = cos(x) + i sin(x),

we can rewrite sin(1/2i) and cos(1/2i) in terms of exponentials.

Therefore, tan(1/2i) = (i(e^(-1/2)) - i(e^(1/2)))/(e^(-1/2) + e^(1/2)) = i(e^(-1/2) - e^(1/2))/(e^(-1/2) + e^(1/2)).