From 52bda5ab3aed6b6d676efe69eb78358815c28745 Mon Sep 17 00:00:00 2001 From: Anthony Berg Date: Wed, 2 Apr 2025 16:55:04 +0200 Subject: [PATCH] Quartz sync: Apr 2, 2025, 4:55 PM --- .../Derivatives/Cauchy-Riemann Equations.md | 9 ++ .../Derivatives/Differentiable.md | 4 + .../Complex Analysis/Derivatives/Entire.md | 4 + .../Derivatives/Holomorphic.md | 8 ++ ...ng 2025-03-24 11.00.27.excalidraw.dark.png | Bin 0 -> 11146 bytes ...ng 2025-03-24 11.00.27.excalidraw.dark.svg | 2 + ...g 2025-03-24 11.00.27.excalidraw.light.png | Bin 0 -> 10887 bytes ...g 2025-03-24 11.00.27.excalidraw.light.svg | 2 + ...ng 2025-03-24 11.13.06.excalidraw.dark.png | Bin 0 -> 21140 bytes ...ng 2025-03-24 11.13.06.excalidraw.dark.svg | 2 + ...g 2025-03-24 11.13.06.excalidraw.light.png | Bin 0 -> 21066 bytes ...g 2025-03-24 11.13.06.excalidraw.light.svg | 2 + ...ng 2025-03-24 11.32.41.excalidraw.dark.png | Bin 0 -> 9728 bytes ...ng 2025-03-24 11.32.41.excalidraw.dark.svg | 2 + ...g 2025-03-24 11.32.41.excalidraw.light.png | Bin 0 -> 9415 bytes ...g 2025-03-24 11.32.41.excalidraw.light.svg | 2 + .../Complex Analysis/Exercises week 1.pdf | Bin 0 -> 111805 bytes content/Lectures/Lecture 19 - Derivatives.md | 129 ++++++++++++++++++ content/index.md | 3 +- 19 files changed, 168 insertions(+), 1 deletion(-) create mode 100644 content/Definitions/Complex Analysis/Derivatives/Cauchy-Riemann Equations.md create mode 100644 content/Definitions/Complex Analysis/Derivatives/Differentiable.md create mode 100644 content/Definitions/Complex Analysis/Derivatives/Entire.md create mode 100644 content/Definitions/Complex Analysis/Derivatives/Holomorphic.md create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.00.27.excalidraw.dark.png create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.00.27.excalidraw.dark.svg create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.00.27.excalidraw.light.png create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.00.27.excalidraw.light.svg create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.13.06.excalidraw.dark.png create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.13.06.excalidraw.dark.svg create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.13.06.excalidraw.light.png create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.13.06.excalidraw.light.svg create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.32.41.excalidraw.dark.png create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.32.41.excalidraw.dark.svg create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.32.41.excalidraw.light.png create mode 100644 content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.32.41.excalidraw.light.svg create mode 100644 content/Exercises/Complex Analysis/Exercises week 1.pdf create mode 100644 content/Lectures/Lecture 19 - Derivatives.md diff --git a/content/Definitions/Complex Analysis/Derivatives/Cauchy-Riemann Equations.md b/content/Definitions/Complex Analysis/Derivatives/Cauchy-Riemann Equations.md new file mode 100644 index 00000000..7545e113 --- /dev/null +++ b/content/Definitions/Complex Analysis/Derivatives/Cauchy-Riemann Equations.md @@ -0,0 +1,9 @@ +# Definition +A [[Complex Functions|complex function]] $f(z)$ can be seen as a function of two real variables. Hence +$$f(z) = f(x + iy) = f(x, y).$$ +When interpreted appropriately. + +> [!example] For instance +> $$f(z) = z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy$$ + +Suppose $f$ is [[Differentiable|differentiable]], we should expect **relations** between $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. \ No newline at end of file diff --git a/content/Definitions/Complex Analysis/Derivatives/Differentiable.md b/content/Definitions/Complex Analysis/Derivatives/Differentiable.md new file mode 100644 index 00000000..84c53d51 --- /dev/null +++ b/content/Definitions/Complex Analysis/Derivatives/Differentiable.md @@ -0,0 +1,4 @@ +# Definition +A function $f : D \to \mathbb{C}$ is **differentiable** at $z \in \mathbb{C}$ if the following limit exists +$$\lim_{ \Delta z \to \infty } \frac{f(z + \Delta z) - f(z)}{\Delta z}.$$ +We denote by $f'(z)$. \ No newline at end of file diff --git a/content/Definitions/Complex Analysis/Derivatives/Entire.md b/content/Definitions/Complex Analysis/Derivatives/Entire.md new file mode 100644 index 00000000..4a56dc05 --- /dev/null +++ b/content/Definitions/Complex Analysis/Derivatives/Entire.md @@ -0,0 +1,4 @@ +# Definition +Let $f: D \to \mathbb{C}$ be a function with domain $D$. + +We say that $f$ is **entire** if $f$ is **entire** if $f$ is [[Differentiable|differentiable]] everywhere (so $D = \mathbb{C}$). \ No newline at end of file diff --git a/content/Definitions/Complex Analysis/Derivatives/Holomorphic.md b/content/Definitions/Complex Analysis/Derivatives/Holomorphic.md new file mode 100644 index 00000000..0f0c71ae --- /dev/null +++ b/content/Definitions/Complex Analysis/Derivatives/Holomorphic.md @@ -0,0 +1,8 @@ +# Definition +Let $f: D \to \mathbb{C}$ be a function with domain $D$. + +We say that $f$ is **holomorphic** at $z_{0}$ if $f$ is [[Differentiable|differentiable]] in an [[Open Sets|open]] disk (entered at $z_{0}$). + +> [!info] Remark on naming of "holomorphic" +> Many sources use the term **analytic** instead of holomorphic. + diff --git a/content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.00.27.excalidraw.dark.png b/content/Excalidraw/Lecture 19/Drawing 2025-03-24 11.00.27.excalidraw.dark.png new file mode 100644 index 0000000000000000000000000000000000000000..7aa7de74d9f0aa6c0185791b651c907dc5baf785 GIT binary patch literal 11146 zcma)ic|4Tw_xG4aA!QFyA^X}$M3%`u46-L{V=wZFRCX%aT9^rA--SXcL_#%om9>(j z?7M7fWO>e{zMtRg_xe7s=lR3yHurtq_jRuKIp=+y<%&9GsI!M*KLY}R*n`#8G(jL} z^5HL!ZYSJ%JZe%57aAWE9d$&>Ti!_of)9b!JZW~-ZZ^&K;X(7NP)mAtihEn*HK`E) z%wu7A!}I8&mvUpd#;do{opYIX#|k(bJIK2)ro>*o_$tBYY}YybHC-9Qb!`9c;WrlD zMA5jMsDuH{-JJLQ-B>EvCz;e3M(bNn%znb%UwYHNTD(=Bj!jlyRhU0Bzf|%fYB$?zC_yNpgpfO%>{3(6aH$kAANQ2Tokp6v=-e zZ~x%^1?$oTvAjF;YWeVY=0aApvOh9XY>l<^B|F{cpw)pX344lOZ?En1%Y{L|erWa1 z^3**&Y3tW!9)EmLnPTMo^JP=c)So8n-AIgb?)`eJ?6Rh)BW`ZeBP<- z%l@?c;&kP_$=205XWu?w%hr_Uo})|~QEh%dzO%N-(vY3FMZdKQ@+B@z>EKqA-Ut5J zcg(kG|4NrvhN)%9B~PJzwh$Y$m|yAAPfj}>Pyq~sq9z<>P|Ml7s&Ew^NuL{x&z(J+ zRsDheHUASKmd>p-5I*{0Q z_1%1sUYhcEbqNc~%bIgb=SmZoa*brP7ssMfY6G92Q7yMgQ~Hiy5oQc3G<8@=wfs1* zNoG$_D;CkiSbTG5X+iEv;-1{%WbquV4Z@z`mmWVRGhBW)p)XAG+5{o-&7y9=;+fnW zrj0(Ak5P`3ZAq?%>p9UETY8*(2OmfJ43TAzv!I<%X_|aD%B-C@>7wu5(;#?MERM)q z_VZ*&qovV`MtNCB*7Wxv#nNE0PRt9MJA zPqz7$3?2_2s95PO^Vr~xe67$4zeom$Qk@i;{f>&1h;C-p z#L4>|!orj@8{X-6BRrocM18Qw_Gr$pacEV~j;|j4vG(;`%0JozI)rz%t8F|93AcZx zE4Ik&&6C1#uN+38g?_${laAo;sF=ty!maoOes-OzzV$&!_(Q2EA-KgyT{Y3M5?eUG zY2451ZGBY4O02J9s`QJ|;LzTsw4On6{FEzMgzpqffqi3l%)vEsPfhChqN?c!XXQ!P zMAFD~NxFN{)}vLSeB&72o60#O>W0ZdDI>|NmCMif=xGyf^NngX!w2EKi~XKNVtd8NmjS`m=4F+Vd{X~fH0ho? 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We have: +$$\frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{z + \Delta z}{\Delta z} = 1 .$$ +Hence $f'(z) = 1$. + + +We have general results to compute derivatives, as in the real case. +# Proposition +Suppose $f$ and $g$ are differentiable at $z \in \mathbb{C}$. Then: +1. $(af(z) + bg(z))' = af'(z) + bg'(z)$ for any $a, b \in \mathbb{C}$, +2. $(f(z)g(z))' = f'(z) g(z) + f(z) g'(z)$, +3. Also a quotient rule. + +Some terminologies which will be important to understand the rest: +- [[Holomorphic]] +- [[Entire]] + +> [!example] +> Any polynomial $P(z) = \sum_{k = 0}^{n} a k z^{k}$ is an [[Entire|entire]] function (follows from the proposition). + +## Example +Consider $f(z) = \; \mid z \mid^{2} = z \bar{z}$. This is not going to be [[Holomorphic|holomorphic]]. + +We compute: +$$\frac{f(z+ \Delta z) - f(z)}{\Delta z} = \frac{\mid z + \Delta z \mid^{2} = \mid z \mid^{2}}{\Delta z}$$ +> [!note] +> We say that $z = x + iy$, hence $\Delta z = a + ib$. + +![[Drawing 2025-03-24 11.00.27.excalidraw.dark.svg]] +%%[[Drawing 2025-03-24 11.00.27.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-24 11.00.27.excalidraw.light.svg|light exported image]]%% + $$= \frac{(x + a)^2 + (y + b)^2 - x^2 - y^2}{a + ib} = \frac{a^2 + b^2 + 2(ax + by)}{a + ib}.$$ +Consider the real direction. + +> [!note] Write +> $$\Delta z = \Delta x$$ + +$$\lim_{ \underbrace{\Delta x \to 0}_{a \to 0} } \frac{f(z + \Delta x) - f(z)}{\Delta x} = \lim_{ a \to 0 } \frac{a^2 + 2ax}{a} = 2x$$ + +Similarly for the imaginary axis +> [!note] Write +> $$\Delta z = i \Delta y$$ + +$$\lim_{ \underbrace{i \Delta y \to 0}_{b \to 0} } \frac{f(z + i \Delta y) - f(z)}{i \Delta y} = \lim_{ b \to 0 } \frac{b^2 + 2by}{ib} = -2iy.$$ +This means that $f$ is **not** [[Differentiable|differentiable]] at $z \neq 0$. +# Cauchy-Riemann Equations +A [[Complex Functions|complex function]] $f(z)$ can be seen as a function of two real variables. Hence +$$f(z) = f(x + iy) = f(x, y).$$ +When interpreted appropriately. + +> [!example] For instance +> $$f(z) = z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy$$ + +Suppose $f$ is [[Differentiable|differentiable]], we should expect **relations** between $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. + +The same thing is also defined here: [[Cauchy-Riemann Equations]] +# Theorem +Let $z_{0} = x_{0} + iy_{0}$. +1. Suppose $f$ is [[Differentiable|differentiable]] at $z_{0}$. Then $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exists and have $$\frac{\partial f}{\partial x}(x_{0}, y_{0}) = -i \frac{\partial f}{\partial y} (x_{0}, y_{0}).$$ +> [!info]- Visual Representation +> +> ![[Drawing 2025-03-24 11.13.06.excalidraw.dark.svg]] +%%[[Drawing 2025-03-24 11.13.06.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-24 11.13.06.excalidraw.light.svg|light exported image]]%% + +2. Suppose $f$ is such that: + - $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exists at $(x_{0}, y_{0})$, + - they are [[Continuous|continuous]] in a small disk centred at $(x_{0}, y_{0})$. + Then $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ satisfy the condition above and $f$ is [[Differentiable|differentiable]] at $z_{0}$. +3. In both cases, we have $f'(z_{0}) = \frac{\partial f}{\partial x}(x_{0}, y_{0})$. +## Proof +### For 1. +Since $f$ is [[Differentiable|differentiable]] at $z_{0}$, the following limits exists +$$f'(z_{0}) = \lim_{ \Delta z \to 0 } \frac{f(z_{0} + \Delta z) - f(z_{0})}{\Delta z}.$$ +> [!info]- Visual representation +> ![[Drawing 2025-03-24 11.32.41.excalidraw.dark.svg]] +%%[[Drawing 2025-03-24 11.32.41.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-24 11.32.41.excalidraw.light.svg|light exported image]]%% + +$$f(z) = f(x, y)$$ + +First we take $\Delta z$ **real**, so $\Delta z = \Delta x$. Then +$$f'(z_{0}) = \lim_{ \Delta x \to 0 } \frac{f(x_{0} + \Delta x) - f(x_{0}, y_{0})}{\Delta x} = \frac{\partial f}{\partial x}(x_{0}, y_{0}).$$ + + +Next take $\Delta z$ to be purely imaginary ($\Delta x, \Delta y \in \mathbb{R}$), so $\Delta z = i \Delta y$ +($z_{0} + i \Delta y = x_{0} + i(y_{0} + \Delta y)$) + +Then: +$$f'(z_{0}) = \lim_{ \Delta y \to 0 } \frac{f(x_{0}, y_{0} + \Delta y) = f(x_{0}, y_{0})}{i \Delta y} = -i \frac{\partial f}{\partial y}(x_{0}, y_{0}).$$ +The two expressions must coincide, since $f$ is [[Differentiable|differentiable]] at $z_{0}$. +### For 2. +It is more complicated to prove. You can find the proof in the references on Canvas for the course. +### For 3. +Shown in [[#For 1.]] +# Decomposition +We can decompose any $f : D \to \mathbb{C}$ into its **real** and **imaginary** part. We write +$$f(z) = f(x, y) = u(x, y) + iv(x, y).$$ +or $\mathrm{Re}f = u$ and $\mathrm{Im} f = v$ +## Corollary +With notation as before, we have +$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\ \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}.$$ +### Proof +By the [[#Theorem|previous theorem]], we have +$$\frac{\partial f}{\partial x} = -i \frac{\partial f}{\partial y}.$$ +Write $f = u + iv$. Then +$$\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = -i \frac {\partial u}{\partial y} + \frac{\partial v}{\partial y}.$$ +Comparing real and imaginary parts gives the result. +## Example +Consider $f(z) = z^2$. +This is seen to be [[Holomorphic|holomorphic]] (in at least two ways). + +One way is +$$f'(z) = (zz)' = z'z + zz' = zz.$$ + +Another way is to use the [[Cauchy-Riemann Equations]]. We have +$$z^2 = (x + iy)^2 = (x^2 - y^2) + 2ixy.$$ +So here +$$\mathrm{Re}f = u = x^2 - y^2,\ \mathrm{Im}f = v = 2xy.$$ +So one checks that $u$ and $v$ satisfy the [[Cauchy-Riemann Equations]]. +Since the derivatives are [[Continuous|continuous]], we get that $f = u + iv$ is [[Holomorphic|holomorphic]]. \ No newline at end of file diff --git a/content/index.md b/content/index.md index 96ea4749..4f9595e3 100644 --- a/content/index.md +++ b/content/index.md @@ -26,4 +26,5 @@ title: ACIT4330 Lecture Notes - [[Lecture 16]] - [[Lecture 17 - Lp Spaces]] # Complex Analysis -- [[Lecture 18 - Complex Analysis]] \ No newline at end of file +- [[Lecture 18 - Complex Analysis]] +- [[Lecture 19 - Derivatives]] \ No newline at end of file