Understanding Functions of Complex Variables
A function of a complex variable maps a complex number ( z = x + yi ) to another unique complex number ( w = u + vi ). Here, both the output's real (( u )) and imaginary (( v )) parts depend on the real (( x )) and imaginary (( y )) parts of the input. For deeper insight into coordinate representations that often underpin these mappings, see Understanding Rectangular and Polar Coordinates for Advanced Function Analysis.
Example: Squaring Function
- ( f(z) = z^2 ) can be expanded as:
(x + yi)^2 = x^2 - y^2 + 2xyi \ - This divides ( f(z) ) into real part ( u = x^2 - y^2 ) and imaginary part ( v = 2xy ), each a function of ( x ) and ( y ).
Holomorphic Functions and Complex Differentiability
A function is holomorphic in a region if it has a unique complex derivative at every point there. Differentiability here follows a similar limit definition as in real calculus:
[ f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} ]
Differentiability in Complex Plane vs Real Line
- Real functions require the left and right derivatives at a point to exist and be equal.
- Complex functions must have derivatives consistent from all directions in the complex plane (infinitely many directions), making the condition much more stringent. This multidirectional approach relates closely to the concepts discussed in Understanding Curvilinear Coordinates: A Comprehensive Guide.
Testing Differentiability: Examples
1. Holomorphic Example: ( f(z) = z^3 )
- Using the limit definition, the derivative is found to be:
f'(z) = 3z^2 \ - Since the limit is independent of the direction of ( \Delta z \to 0 ), ( z^3 ) is holomorphic.
- In general, ( f(z) = z^n ) (with ( n ) positive integer) is holomorphic with derivative ( nf(z) = nz^{n-1} ).
2. Non-Holomorphic Example: ( f(z) = 2y + xi )
- Analyzing the derivative limits along the real axis (( \Delta y=0 )) and imaginary axis (( \Delta x=0 )) yields different results:
- Along ( x )-axis: derivative limit ( = i )
- Along ( y )-axis: derivative limit ( = -2i )
- Since limits differ depending on approach direction, ( f(z) ) is not holomorphic.
Summary and Next Steps
- Complex differentiability requires directional independence of the derivative, leading to powerful results in complex analysis.
- Holomorphic functions enable tools like contour integration and have differentiable real and imaginary parts satisfying specific conditions.
- Upcoming lectures will explore the Cauchy-Riemann equations, essential for characterizing holomorphic functions in detail, foundational for Understanding First-Order Ordinary Differential Equations: Geometric and Numerical Approaches.
What makes a complex function holomorphic?
- It must have a unique complex derivative at every point in its domain, independent of the approach direction.
Why is this stricter than real differentiation?
- Because complex numbers approach a point from infinitely many directions on the plane rather than just two on the line.
How to verify holomorphicity?
- Use the limit definition of the derivative considering all directions, or apply the Cauchy-Riemann conditions (to be covered).
This foundational understanding sets the stage for advanced topics in complex calculus and mathematical physics applications.
welcome back everyone to my lecture series on mathematical physics today's topic is going to be an introduction to
functions of complex variables there's some really fun and Powerful stuff here so get ready let's start with the really
basic stuff what do I mean by a function of a complex variable put simply a function of a complex variable is a
relation that Maps a complex number Z to another unique complex number now because the input Z is composed of a
real part and an imaginary part like any complex number the output W which is also a complex number is also composed
of a real and imaginary part note that I've used I to denote the imaginary number
here generally the real and imaginary parts of w will themselves be functions of the real and imaginary parts of Z in
other words u and v will both be functions of X and Y let's try to make sense of this with an example suppose
that my complex function is just z^2 I can plug in my Z which is x + Yi to rewrite this function as x + Yi s which
is x + y i * x + Yi I can expand this out to get x^2 + 2xy I + y i and then use the fact that i^2 = -1 to
write this expression as x^2 - y^ 2 + 2xy I so now my complex function f of Z has been broken up into a real part U
and an imaginary part V which is exactly what we said earlier that you can break up a complex function into real and
imaginary Parts which can then be written as functions of the real and imaginary parts of the input VAR
variable Z so just like how a complex number Z is a pair of real numbers put together a complex function f is a pair
of real functions put together eventually our goal with complex functions is to do calculus on them so
things like differentiation and integration specifically Contour integration but in order to do all that
we need our complex functions to be differentiable to have their derivatives defined and continuous this leads us
nicely to the concept of holomorphic functions a function f of Z is holomorphic in a region R of the complex
plane if it has a unique derivative at every point in R what do I mean by derivative how's the derivative defined
for complex functions well it's defined the same way as it is for Real functions in other
words the derivative of f of Z which is fime z or DF DZ if you prefer liveness notation it's the limit of Delta F which
is a change in F over Delta Z which is a change in Z as Delta Z approaches zero now here's where the connection between
real functions and complex functions starts coming up for a complex function to be holomorphic or analytic or regular
whichever term you like to use the conditions that need to be met are much more stringent than the conditions for
real functions recall that if you had a real function f ofx X for f ofx to be
differentiable at some point a you needed the limit of f of a plus hus F of a over H as H approaches zero needed
that limit to exist notice that I'm using H to denote a small change in X and I've done the same for Z as well so
hopefully that's not too confusing for this limit to exist we need the limit from the right and the
limit from the left to exist additionally we need those two limits to equal each other in other words the
function has to have its right derivative and left derivative equal each other at that point if they weren't
equal as is the case of a function that has a corner like this then the function would not be differentiable at that
point because its left derivative and right derivative are both different so that limit as H approaches zero doesn't
exist because it has two unique values depending on the direction in which uh your H approach is zero for a complex
function however the Criterion is much more stringent purely because of the nature of complex numbers for a real
function the real number X that serves as the input of the function that real number can only approach a given value a
from two directions from the left and from the right because X exists only on a number line however a complex number
doesn't just exist on a number line it exists on a whole plane so it can approach a value a from above low left
right from any diagonal Direction so a complex function has to be left differentiable right differentiable up
differentiable down differentiable and diagonally differentiable and it has to have its derivatives from all those
directions equal each other for it to be considered differentiable at a given point a so clearly this is a pretty
strict requirement but because this requirement is so strict once we satisfy it we'll have some really powerful
results but before we do that let's look at some examples of a differentiable and a non-differentiable complex
function our first example will be a simple cubic function to check whether or not it's differentiable let's just
apply the definition of the derivative which is DF DZ is the limit as Delta Z approaches Z of Delta F over Delta Z
which is the same as the limit as H approaches Z of f of z+ hus F of z h I'm going to Omit writing the limits every
time just to make things easy for myself anyway F of Z Plus H is just z+ H cubed and F of Z is just Z Cub so we can plug
that in and then expand out the Z Plus H term now the Z Cub terms in the numerator cancel out and we end up with
3 z^2 H + 3 Z h^ 2+ HB all over H now the H's in the numerator cancel out with the H's in the denominator and we end up
with a simpler expression that's 3 Z ^2 + 3 z h plus plus h^2 we're not done yet because we still have to take the limit
as H approaches zero and if we do that we'll just get 3 z^2 since these other two terms are going to cancel out now in
this particular example the direction along which Delta Z or which H approach zero is irrelevant it could approach
from any direction and it wouldn't matter because it ends up canceling out in the end
so from this fact we can conclude that z cubed is a holomorphic function and in general we can say that a complex
function Z to the N is holomorphic and its derivative is just n * Z the N minus1 where N is a positive integer the
other example we'll be doing is f of Z which is U + VI is 2 y + x i again let's use the definition of the derivative the
only difference is that now we have a function in terms of X and Y instead of just Z so to make things easier we'll
use Delta Z instead of the H we used earlier because Z itself is just x + Yi it follows that Delta Z is just Delta x
+ Delta y * I so F of Z + Delta Z is 2 * y + Delta y + x + Delta x * I subtracting f Z from this expression
gives us 2 Delta y+ Delta x * I and if we plug this into the definition of the derivative we'll get the limit as Delta
Z approaches Z of 2 Delta y+ Delta x i over Delta x + Delta y * I now here the manner by which we make
Delta Z approach zero changes the value of the derivative we get for instance if we approach zero along or parallel to
the x axis so if we approach it from either the left or the right that means Delta Y is always going to be zero since
only X is changing so we're left with the limit as Delta Z approaches zero of Delta x * I over Delta X and that just
equals I however if we approach zero along or parallel to the Y AIS we approach it
vertically that means Delta X is always going to be zero and so we're left with the limit as Delta Z approaches Z of 2 2
Delta y/ Delta y * I which is 2 I which is -2 I now clearly these two derivatives one gives you I the other
gives you -2 I they're different but for a function to be holomorphic the derivatives along all directions have to
exist and they have to equal each other so we conclude that the function f of Z which is 2 y + x i is not holomorphic so
even though F of Z here is individually composed of two differentiable functions in real Space 2 Y and X they're both
differentiable in real space the complex function f of Z overall is not complex differentiable it's not holomorphic and
that wraps it up for this video in the next video we're going to be going further with holomorphic functions and
we're going to start looking at the CI Ray modulations
A function of a complex variable maps a complex number ( z = x + yi ) to another complex number ( w = u + vi ), where both the real part (( u )) and imaginary part (( v )) depend on the input's real (( x )) and imaginary (( y )) parts. This representation allows analyzing complex functions in terms of their component real-valued functions of two variables.
A complex function is holomorphic in a region if it has a unique complex derivative at every point there, meaning the limit ( f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} ) exists and is independent of the direction from which ( \Delta z ) approaches zero. Practically, verifying this involves checking that the derivative limits are the same from all directions or using the Cauchy-Riemann equations for a more straightforward test.
In real calculus, derivatives must be consistent from two directions—left and right. However, in complex analysis, derivatives must be consistent from infinitely many directions in the complex plane, not just along a line. This multidirectional requirement makes holomorphicity a stronger and more restrictive condition than real differentiability.
Yes, for example, the function ( f(z) = z^3 ) is holomorphic everywhere. Its derivative, found using the limit definition, is ( f'(z) = 3z^2 ). Because the limit defining the derivative is independent of the direction of approach, ( z^3 ) satisfies holomorphicity. More generally, ( f(z) = z^n ) with a positive integer ( n ) is holomorphic with derivative ( nf(z) = nz^{n-1} ).
The function ( f(z) = 2y + xi ) (where ( z=x+yi )) is not holomorphic because its derivative limits differ depending on the direction from which ( \Delta z ) approaches zero. Specifically, along the real axis, the derivative limit is ( i ), while along the imaginary axis, it is ( -2i ). This directional dependence means the complex derivative does not exist uniquely, violating holomorphicity.
Holomorphic functions have powerful properties, including being infinitely differentiable and conforming to the Cauchy-Riemann equations. They enable advanced techniques like contour integration and play a crucial role in complex analysis and applications in physics and engineering. Verifying holomorphicity is often a starting point for deeper explorations of complex function behavior.
After grasping holomorphicity, the next key topic is the Cauchy-Riemann equations, which give necessary and sufficient conditions for a function to be holomorphic. Studying these equations helps characterize complex differentiability more easily and forms the foundation for advanced topics such as contour integration, complex integration theorems, and applications in differential equations and mathematical physics.
Heads up!
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