Differential Calculus Problem Help w/ Complex Numbers

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SUMMARY

This discussion focuses on solving differential calculus problems involving complex numbers, specifically the interpretation of regions defined by inequalities such as |z|<2. The user seeks clarification on sketching these regions and approaches to specific sections of their calculus problems. Key concepts mentioned include the modulus-argument form of complex numbers, Euler's identity, and the multivalued nature of the complex exponential function. The user is encouraged to express complex numbers in the form z=x+iy to facilitate understanding.

PREREQUISITES
  • Understanding of complex numbers and their representation
  • Familiarity with differential calculus concepts
  • Knowledge of Euler's identity and its applications
  • Ability to interpret inequalities in the complex plane
NEXT STEPS
  • Study the modulus-argument form of complex numbers in detail
  • Learn how to sketch regions in the complex plane based on inequalities
  • Explore applications of Euler's identity in calculus
  • Investigate the properties of multivalued functions in complex analysis
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Students and educators in mathematics, particularly those studying complex analysis and differential calculus, as well as anyone seeking to deepen their understanding of complex number applications in calculus.

alexialight
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Just wondering if anyone could help me out with some problems I'm having with differential calculus.

Firstly, can anyone confirm if I sketched the region in image 1.jpg correctly (shown in 2.jpg)? I've done questions before where it just says |z|<2 and I know that it looks like a circle, but in this case the Re z at the end of the 2 just got me really confused.

Also, in image 3.jpg I have no idea as to how to even approach section a and I'm pretty sure I can't do b and c without understanding a. Can anyone help me out with this? Any hints or something because I've been staring at it for a while now and not getting anywhere (obviously).
 

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For the first one, maybe writing z=x+iy (with Re(z)=x) will help.

For the second one, z=r(cost + isint) is called the modulus-argument form, i.e. r and t are the modulus and argument of z, respectively.

Does that help?
 
For the problem,make use of Euler's identity

[tex]e^{i\varphi}\equiv \cos\varphi+i\sin\varphi[/tex]

and of the multivaluedness of the complex exponential.

Daniel.
 

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