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Firepanda
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I'm not too sure how to show this. Perhaps if I show d(ez)/dz = ez then does this conclude that ez is holomorphic on all of C?
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Dick said:Why does d/dz(e^z)=e^z show e^z is holomorphic? Or at least, why are you saying you aren't sure this shows it's holomorphic?
Firepanda said:I suppose I should be using the cauchy riemann equations then?
I was told not to use limits here so I didn't want to use the base definition of holomorphic-ness.
Firepanda said:i tried to use C-R, but I'm unable to split it up into
U(x,y) or V(x,y)
using z=x+iy into cosh(z)
Dick said:How about e^z. Can you split that up?
Sure, check e^(-z) as well. It's almost the same thing. I'm a little puzzled why you can split e^(z) and e^(-z) up and not cosh(z). You already said cosh(z)=(e^z+e^(-z))/2.Firepanda said:exeiy
= ex(cosy + i.siny)
=excosy + i.exsiny
The C-R satisfy this and so it is holomorphic on all of C? so this would immediatly imply cosh(z) is entire?
And I would check if it were true for e-z as well.
Dick said:Sure, check e^(-z) as well. It's almost the same thing. I'm a little puzzled why you can split e^(z) and e^(-z) up and not cosh(z). You already said cosh(z)=(e^z+e^(-z))/2.
A holomorphic function is a complex-valued function that is defined and differentiable on an open set in the complex plane. It is also known as an analytic function, as it can be represented by a convergent power series.
A holomorphic function is a function that is defined and differentiable on a complex plane, while a regular function can be defined and differentiable on a real plane. Additionally, a holomorphic function satisfies the Cauchy-Riemann equations, which relate the real and imaginary parts of the function, while a regular function does not have this property.
Some examples of holomorphic functions include polynomials, exponential functions, and trigonometric functions. Any function that can be represented by a convergent power series is also a holomorphic function.
Holomorphic functions have many applications in mathematics and science, particularly in complex analysis, which studies the properties and behavior of functions on the complex plane. They are also used in physics, engineering, and other fields to model and solve problems involving complex variables.
Holomorphic functions are used in many real-world applications, such as in signal processing, image analysis, and financial modelling. They are also used in computer graphics, as they can be used to represent and manipulate complex shapes and surfaces. In addition, holomorphic functions are used in the design and analysis of electronic circuits and systems.