If `0ltargzlepi//4` find the least vlaue of `sqrt2|z-1|`

Answer

To find the least value of \( \sqrt{2} |z - 1| \) given that \( 0 < \arg z \leq \frac{\pi}{4} \), we can follow these steps:

### Step 1: Understand the Argument Condition
The condition \( 0 < \arg z \leq \frac{\pi}{4} \) indicates that the complex number \( z \) lies in the first quadrant between the positive x-axis and the line \( y = x \).

### Step 2: Express \( z \) in Terms of Cartesian Coordinates
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The argument of \( z \) can be expressed as:
\[ \arg z = \tan^{-1}\left(\frac{y}{x}\right) \]
Given the argument condition, we have:
\[ 0 < \frac{y}{x} \leq 1 \implies 0 < y \leq x \]

### Step 3: Find the Expression for \( |z - 1| \)
We need to calculate \( |z - 1| \):
\[ |z - 1| = |(x - 1) + iy| = \sqrt{(x - 1)^2 + y^2} \]

### Step 4: Substitute \( y \) in Terms of \( x \)
Since \( y \leq x \), we can express \( y \) as \( y = kx \) where \( 0 < k \leq 1 \). Thus, we can rewrite \( |z - 1| \):
\[ |z - 1| = \sqrt{(x - 1)^2 + (kx)^2} = \sqrt{(x - 1)^2 + k^2x^2} \]

### Step 5: Minimize \( \sqrt{2} |z - 1| \)
We want to minimize:
\[ \sqrt{2} |z - 1| = \sqrt{2} \sqrt{(x - 1)^2 + k^2x^2} \]
This simplifies to:
\[ \sqrt{2} \sqrt{(x - 1)^2 + k^2x^2} = \sqrt{2((x - 1)^2 + k^2x^2)} \]

### Step 6: Analyze the Function
To find the minimum value, we can analyze the function geometrically. The point \( (1, 0) \) represents the complex number \( 1 \) on the complex plane, and we want to find the perpendicular distance from the line \( y = x \) to the point \( (1, 0) \).

### Step 7: Calculate the Perpendicular Distance
The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( y = x \) (or \( x - y = 0 \)), we have \( A = 1, B = -1, C = 0 \), and the point \( (1, 0) \):
\[ d = \frac{|1 \cdot 1 + (-1) \cdot 0 + 0|}{\sqrt{1^2 + (-1)^2}} = \frac{1}{\sqrt{2}} \]

### Step 8: Final Calculation
Now substituting back into our expression:
\[ \sqrt{2} |z - 1| = \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 \]

Thus, the least value of \( \sqrt{2} |z - 1| \) is:
\[ \boxed{1} \]