Prove that if φis real, then Re(e^i φ z) ≤|z|, with equality if and only if φ=-ph z.

Name: Problem 9, Chapter 3: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2020-09-21T13:36:29-08:00
Duration: 4 min 55 s
Channel: John Gehad
Description: Prove that if φis real, then Re(e^i φ z) ≤|z|, with equality if and only if φ=-ph z.

step 1 Problem number 74, R equals x, y and z. Therefore, the curl of 1 over the norm of r power p as t

Prove that if φis real, then Re(e^i φ z) ≤|z|, with equality...

Key Concepts

Argument and Conditions for Equality

The argument (or phase) of a complex number indicates its angular direction in the complex plane. Equality in the inequality is achieved precisely when the cosine of the corresponding angle equals 1, meaning the complex number is aligned on the real axis after rotation. This occurs when the rotation precisely cancels the original argument of the complex number.

Inequalities Involving Cosine

A fundamental property of the cosine function is that |cos ?| is always less than or equal to 1. This property is used to show that the real part of a rotated complex number, which is |z|cos?, cannot exceed |z|, forming the basis of the inequality.

Complex Numbers and Polar Representation

Every complex number can be expressed in polar form as |z|e^(i?), where |z| represents the modulus (or magnitude) and ? represents the argument (or phase). This form is essential in understanding how complex numbers behave under multiplication, particularly in terms of rotations in the complex plane.

Euler's Formula and Complex Exponentials

Euler's formula, which states that e^(i?) = cos? + i sin?, provides a bridge between exponential and trigonometric representations of complex numbers. Through this formula, multiplying by e^(i?) is interpreted as a rotation of a complex number by an angle ?, an idea central to many results in complex analysis.

Real Part of a Complex Number

The real part of a complex number is defined as the component along the real axis. When a complex number is rotated by an angle via multiplication with e^(i?), its real part becomes the projection of the rotated vector along the real axis, which is governed by the cosine of the adjusted angle.

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