Triangle Inequality in Complex Analysis

Triangle Inequality in Complex Analysis

What is the Triangle Inequality?

In Complex Analysis, it states that any two complex number z1 and z2, the equality holds:

|z1 + z2| ≤ |z1| + |z2|


Here, |z| denotes the modulus of the complex number, and the modulus of complex number z = x + yi, where x is the real number and yi is the imaginary number. So, in complex analysis is defined as,

|z| = √(x2 + y2)

What is the Triangle Inequality number theory?

In Number Theory, the Triangle Inequality is discussed in the context of the absolute value function over the integers or the real number. It states that for any real number a and b, the Inequality holds:

What is the triangle inequality modulus formula?

As we discuess above,

|z1 + z2| ≤ |z1| + |z2|

Proof of Triangle Inequality in Complex Analysis

If z1 and z2 are two complex numbers, the prove that
|z1 + z2| ≤ |z1| + |z2|
Statement. |z1 + z2|2 = (z1 + z2)

If \( z_1 \) and \( z_2 \) are two complex numbers, prove that

\( |z_1 + z_2| \leq |z_1| + |z_2| \)

Proof:

\( |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = (z_1 + z_2)(\overline{z_1} + \overline{z_2}) \)

\( |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = (z_1 + z_2)(\overline{z_1} + \overline{z_2}) \)

\( = z_1\overline{z_1} + z_2\overline{z_2} + z_1\overline{z_2} + \overline{z_1}z_2 \)

\( = |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + \overline{(z_1\overline{z_2})} \)

[∵ \( \overline{(z_1\overline{z_2})} = \overline{z_1}(\overline{\overline{z_2}}) = \overline{z_1}z_2 \) ]

\( = |z_1|^2 + |z_2|^2 + 2\operatorname{Re}(z_1\overline{z_2}) \)

[∵ \( \operatorname{Re}(z) = \frac{z + \overline{z}}{2} \), i.e., \( 2\operatorname{Re}(z) = z + \overline{z} \) ]

\( \leq |z_1|^2 + |z_2|^2 + 2 |z_1||z_2| \)

[∵ \( x^2 \leq x^2 + y^2 \), ∴ \( |x|^2 \leq |z|^2 \) or \( |x| \leq |z| \), i.e., \( \operatorname{Re} z \leq |z| \) ]

\( = |z_1|^2 + |z_2|^2 + 2 |z_1||\overline{z_2}| = |z_1|^2 + |z_2|^2 + 2 |z_1||z_2| \)

[∵ \( |\overline{z}| = |z| \) ]

\( = (|z_1| + |z_2|)^2 \)

∴ \( |z_1 + z_2|^2 \leq (|z_1| + |z_2|)^2 \)

∴ \( |z_1 + z_2| \leq |z_1| + |z_2| \)

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