Introduction to Holomorphic Functions
A complex function is holomorphic if it is differentiable everywhere in its domain. This requires special relationships between its real and imaginary parts. For a detailed overview, see Introduction to Functions of Complex Variables and Holomorphicity.
Cauchy-Riemann Relations
- Holomorphicity requires the real part (u) and imaginary part (v) of a complex function f(z) = u + iv to satisfy the Cauchy-Riemann equations:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
- These relations are necessary conditions for holomorphicity.
Theorems on Holomorphicity
- Necessary Conditions: If a function is holomorphic, it must satisfy the Cauchy-Riemann equations.
- Sufficient Conditions: If u and v have continuous partial derivatives in a region and satisfy the Cauchy-Riemann equations there, the function is holomorphic in that region.
Example 1: Function z3
- Real part u and imaginary part v are polynomials with continuous derivatives.
- Verified that the Cauchy-Riemann relations hold:
- ∂u/∂x = ∂v/∂y = 3x2 - 3y2
- ∂u/∂y = -6xy, ∂v/∂x = 6xy (negatives of each other)
- Conclusion: z3 is holomorphic everywhere.
Example 2: Function 2y + xi
- Both u = 2y and v = x are continuous with continuous derivatives.
- First Cauchy-Riemann relation holds: ∂u/∂x = 0, ∂v/∂y = 0
- Second relation fails: ∂u/∂y = 2, ∂v/∂x = 1 (not negatives)
- Conclusion: Function is not holomorphic anywhere.
Additional Key Theorems
- Infinite Differentiability: A holomorphic function is infinitely differentiable within its domain.
- Taylor Series Expansion: Any holomorphic function can be expressed as a Taylor series within a radius defined by the nearest singularity (point where the function is not holomorphic).
- For complex functions, the radius of convergence forms a circular region in the complex plane.
Harmonicity of Real and Imaginary Parts
- Real and imaginary parts of holomorphic functions satisfy Laplace's equation, making them harmonic functions.
- Any harmonic function in a suitable region can be the real or imaginary part of a holomorphic function.
Summary
- Holomorphic functions satisfy the Cauchy-Riemann equations with continuous partial derivatives.
- These conditions ensure powerful properties like infinite differentiability and representability by Taylor series.
- The distinction between holomorphic and non-holomorphic functions can be demonstrated by checking these relations on practical examples.
This foundational understanding is crucial for further exploration and applications in complex analysis and mathematical physics. To deepen your study, consider exploring Understanding Rectangular and Polar Coordinates for Advanced Function Analysis, which complements the coordinate framework for complex functions.
in the last video we looked at two examples of complex functions one of which was holomorphic while the other
wasn't holomorphic by holomorphic I mean it was differentiable for these examples Z Cub was holomorphic or differentiable
everywhere while 2 y + x I wasn't holomorphic anywhere so how is it that a function with these real and imaginary
Parts was holomorphic but a function with these real and imaginary Parts wasn't holomorphic it's because for a
complex function to be holomorphic we need some special relationships between its real and imaginary parts to be
satisfied so even though two Y and X are each continuous and differentiable in real space the fact that they don't
satisfy these special relationships as real and imaginary Parts makes the complex function overall non-holomorphic
the relationships between the real and imaginary parts of a complex function that need to be satisfied for it to be
Hollow orphic are called the CI raymon relations in fact there's two theorems that attest to this the first theorem
just says that if I have a holomorphic complex function something like this then its real and imaginary Parts
satisfy du by DX equals DV by Dy and the second relationship is that du by Dy equals DB by DX and these relationships
are satisfied in the region r where the complex function is holomorphic these two equations or
relationships are called the CI raymon relations all this first theorem says is that the Ki raymon relations are
necessary conditions for holomorphicity doesn't say anything about them being sufficient in other words if a function
doesn't obey the Ki raymon relations then it can't be holomorphic the second theorem however
does talk about sufficient conditions this guy states that if the following condition are met in a region R of the
complex plane the first is that the partial derivatives of u and v with respect to X and Y exist the second is
that they're continuous and the third is that they satisfy the CI Ray modulations if all these three conditions are met
then the function f of Z which is composed of the real part U and the imaginary part V that function is Hollow
morphic in the region R I won't be proving any of these two theorems unless everyone gets pissed at me in which case
I will make a video but in the context of a mathematical physics lecture series I think stating these theorems should
suffice now that we know about the Ki raymon relations we can go back to our two examples and verify using Ki Ron why
one is holomorphic and the other Isn't So if I copy this from up here there we go so let's verify that z
cubed is holomorphic using the second theorem well the the real and imaginary Parts u and v are both pols so clearly
their partial derivatives must exist and must be continuous so that's not a problem but are the Ki raymon relations
satisfied let's check the partial derivative of U with respect to X is just bring the power down and then
reduce it by one 3x^2 - 3 y^2 because remember Y is just a constant when we're taking the partial derivative with
respect to X the partial derivative of V with respect to Y is also 3x^2 - 3 y^2 and now X is constant in this case if we
go back to the CI raymon relations we know that DX must equal DV Dy and in this case that's true so the first
relation Works let's check the second relation partial U partial Y is just -6x y while partial V partial X is just
6xy from the Ki raymon relations these guys have to be negatives of each other and in this case that's true so we
verified using the second theorem that z cubed is indeed a holomorphic function now let's verify that the
second example 2 y + x i is not holomorphic again 2 Y and X are both continuous and have continuous
derivatives so that's not an issue which means all we have left is to check the Ki Ray
modulations here Dux is zero because there's no X in this term and DV Dy is also zero because there's no Y in this
term since these two have to be equal we've actually verified that the first CI raymon relation holds however du Dy
is just two and the partial V partial X is just one since these two have to be negatives of each other the second CI
raymon relation isn't satisfied because of this and because of the fact that from the first theorem the Ki raymon
relations have to be satisfied for a function to be holomorphic we can conclude that 2 y + XI is not
holomorphic which is exactly what we found in the last video albe it with a different approach there's a couple of
more theorems left none of which I'll prove sadly before I end this video one of these theorems is a really powerful
one what it states is that if I have a complex function that's holomorphic or differentiable in a region r on the
complex plane then that function is infinitely differentiable in the region R so if the first derivative of my
complex function exists and it's continuous then it follows that derivatives of all orders of that
function exist and are continuous that's really powerful stuff another theorem has to do with tailor series expansions
of complex functions so again if I have a function that's holomorphic in R then I can write that function as a tailor
series around any point in R this is because if it's holomorphic then I know from the previous theorem that it's
infinitely differentiable which is why I can write a tailor series for it because remember when you want to write a tailor
series for a function of a single variable you need that function to be infinitely differentiable about the
point where you're expanding and this is what this theorem makes use of the radius of convergence of that tailor
series or power series will be the distance between Z KN which is the Point you're expanding about and the singular
Point that's closest to Z which is why I have a minimum in this expression by singular Point what I mean is a point
where F of Z is not complex differentiable it's not holomorphic it's derivative isn't defined so let's
illustrate this idea of radius of convergence of a tailor series of a complex function say I have a complex
function 1 over Z we can clearly see that f is going to be undefined at Z equals 0 so although I can't make a
tailor series expansion around zal 0 I might be able to make one at say zal 1 what this theorem says is that when I
make a tailor expansion of 1 / Z about zal 1 then the region in which my tailor expansion works is a circle centered at
zal 1 with a radius of one note that because complex numbers exist on a plane the region of convergence is now a
circle instead of a mere interval because remember in real functions you had an interval of convergence but for
complex functions you have a region of convergence because complex numbers are effectively two-dimensional numbers I'm
going to State One Last Theorem before ending this video this theorem states that if I have a holomorphic function f
then it's real and imaginary Parts both satisfy lass's equation in the region where f is holomorphic it's actually
very easy dare I say trivial to prove this just by differentiating the Ki raymon relations and doing a little bit
of algebra so because they satisfy lass's equation the real and imaginary parts of a
complex function u and v are considered harmonic functions in fact any function that satisfies lass's equation in a nice
enough region can possibly be the real or imaginary part of a holomorphic complex function anyway that does it for
this video uh but before I go let me apologize for the fact that for most of this video I chucked a whole bunch of
theorems at you without really proving anything if theorems intimidate you then I'm sorry but for me it's kind of
unavoidable to use them when teaching complex variables but I promise that after I'm done introducing the basic
concepts I'll do my best to go over some applications and problems thanks for watching
The Cauchy-Riemann equations are a set of two partial differential equations that relate the real part (u) and imaginary part (v) of a complex function f(z) = u + iv. Specifically, they require ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. These equations are necessary for a function to be holomorphic, meaning complex differentiable everywhere in its domain, and ensure the function's differentiability in the complex sense.
To verify holomorphicity, separate the complex function into its real part u(x,y) and imaginary part v(x,y). Then compute the partial derivatives ∂u/∂x, ∂u/∂y, ∂v/∂x, and ∂v/∂y. Check if the Cauchy-Riemann equations ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x hold throughout the domain, and ensure these derivatives are continuous. If these conditions are met, the function is holomorphic in that region.
Yes. For example, the function f(z) = z^3, where z = x + iy, can be expressed with real part u = x^3 - 3xy^2 and imaginary part v = 3x^2y - y^3. These parts satisfy the Cauchy-Riemann equations with continuous derivatives, so z^3 is holomorphic everywhere. Conversely, the function f(z) = 2y + xi has u = 2y and v = x, which do not satisfy the second Cauchy-Riemann equation, hence this function is not holomorphic anywhere.
Holomorphic functions are infinitely differentiable and can be expressed as Taylor series expansions within a radius defined by the nearest singularity. This radius of convergence corresponds to a circular region in the complex plane. These powerful properties stem from holomorphicity and enable advanced analysis and applications in complex variable theory.
The real and imaginary parts of a holomorphic function satisfy Laplace's equation and are thus harmonic functions. This means each part exhibits properties like mean value and maximum principles. Moreover, any harmonic function defined in a suitable region can be the real or imaginary component of some holomorphic function, highlighting a deep link between harmonic and holomorphic functions.
The necessary condition is that the real and imaginary parts satisfy the Cauchy-Riemann equations. The sufficient condition requires that these parts have continuous partial derivatives in the region of interest and satisfy the Cauchy-Riemann equations there. When both conditions hold, the function is holomorphic within that domain.
Heads up!
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