Complex Numbers

To solve the problem, we need to evaluate the infinite product of the complex numbers given by:

1. Recognizing the Euler Form:
The expression is of the form:

So, we rewrite:

2. Evaluate the Infinite Product:
We are asked to compute:

3. Use Product of Exponentials:

4. Evaluate the Geometric Series:

5. Final Expression:

Final Answer:
The value of the infinite product is .

To solve the problem, we need to find the locus of the complex number that satisfies:

1. Let :
Then, , and

2. Express the modulus condition:

3. Use modulus of a complex number:
, so:

4. Square both sides:

5. Expand the squares:
Numerator:
Denominator:

6. Plug into equation:

7. Cross-multiply:

8. Expand both sides:
LHS:
RHS:

9. Bring all terms to one side:

Combine like terms:

10. Multiply both sides by -1:

Final Answer:
The locus of is given by the equation:

To solve the problem, we need to find the difference in arguments (angles) of the roots of the complex equation .

1. Solve the Equation:
We are given the equation:

To find , we take square roots of :
In polar form, , so:

where , .
So the roots are:

2. Find Arguments of Roots:

3. Calculate the Difference:

Final Answer:
The value of is .

Step 1: Simplify and First, compute each part: Expanding and simplifying, we use : For the denominator: So, it simplifies to:

Step 2: Simplify the expression for Now, plug these into the expression for : Expanding the numerator: Then dividing by the denominator:

Step 3: Calculate the argument of To find , use the tangent of the angle: Thus, . Since the real part is negative and the imaginary part is positive, is in the second quadrant. Therefore, the principal value of is: However, using the identity: We adjust to be in the correct format given in the options: This is consistent with option (1), reflecting the negative real component and positive imaginary component.

Step 1: Expressing in exponential form.
Since , we express them as and .

Step 2: Evaluating .

Step 3: Conclusion.
Thus, .