MCC201A — Formula Sheet

• $z = x + iy = r(\cos\theta + i\sin\theta) = r\,e^{i\theta}$, with $r = |z| = \sqrt{x^2 + y^2}$
• Principal argument $\operatorname{Arg}(z) \in (-\pi, \pi]$: in Q1/Q4, $\arctan(y/x)$; in Q2, $\arctan(y/x) + \pi$; in Q3, $\arctan(y/x) - \pi$
• Multiplication: $z_1 z_2 = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$ — moduli multiply, arguments add
• De Moivre: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$, $\ n \in \mathbb{Z}$
• $n$-th roots of $w = R\,e^{i\varphi}$: $z_k = R^{1/n}\,e^{\,i(\varphi + 2k\pi)/n}$, $\ k = 0, \dots, n-1$
• $n$-th roots of unity $\omega_k = e^{2k\pi i/n}$ sum to zero: $\sum_{k=0}^{n-1}\omega_k = 0$

• $f(z) = u(x,y) + i\,v(x,y)$ — one complex function is two real ones
• Limit: $\lim_{z\to z_0} f(z) = L$ must give the same value along every path; $\varepsilon$-$\delta$ form $\ 0<|z-z_0|<\delta \Rightarrow |f(z)-L|<\varepsilon$
• Split rule: $\lim f = \lim u + i\lim v$ — both real limits must exist
• Two-path test: two paths with different limits $\Rightarrow$ the limit does not exist
• Continuous at $z_0$: $\lim_{z\to z_0} f(z) = f(z_0)$
• $f'(z_0) = \lim_{h\to 0}\dfrac{f(z_0+h) - f(z_0)}{h}$, same value from every direction of $h$; differentiable $\Rightarrow$ continuous

• CR equations: $\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}$ and $\dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}$
• Sufficient test: $u,v$ have continuous first partials and CR holds throughout a region $\Rightarrow f$ analytic there
• $f'(z) = \dfrac{\partial u}{\partial x} + i\dfrac{\partial v}{\partial x} = \dfrac{\partial v}{\partial y} - i\dfrac{\partial u}{\partial y}$
• Polar form: $\dfrac{\partial u}{\partial r} = \dfrac{1}{r}\dfrac{\partial v}{\partial\theta}$, $\ \dfrac{\partial v}{\partial r} = -\dfrac{1}{r}\dfrac{\partial u}{\partial\theta}$
• $u$ and $v$ are harmonic: $\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0$
• Differentiable at one isolated point $\ne$ analytic — analyticity needs a whole region

• $v$ is a harmonic conjugate of $u$ when $u + iv$ is analytic; on a connected domain, unique up to an additive real constant
• Direct method: $dv = -\dfrac{\partial u}{\partial y}\,dx + \dfrac{\partial u}{\partial x}\,dy$, then $f(z) = u + iv$ re-expressed in $z$
• Milne-Thomson (given $u$): $f'(z) = \dfrac{\partial u}{\partial x} - i\dfrac{\partial u}{\partial y}$, then set $x = z,\ y = 0$; integrate to $f(z) = F(z) + iC,\ C \in \mathbb{R}$
• Milne-Thomson (given $v$): $f'(z) = \dfrac{\partial v}{\partial y} + i\dfrac{\partial v}{\partial x}$, then set $x = z,\ y = 0$; integrate to $f(z) = F(z) + C,\ C \in \mathbb{R}$
• Orthogonal families: $\nabla u \cdot \nabla v = 0$ for analytic $f = u + iv$ — level curves $u = c_1$ and $v = c_2$ meet at right angles wherever $f'(z) \ne 0$
• $e^z = e^x(\cos y + i\sin y)$; $\quad \sin z = \dfrac{e^{iz} - e^{-iz}}{2i}$; $\quad \cos z = \dfrac{e^{iz} + e^{-iz}}{2}$
• $\sinh z = \dfrac{e^{z} - e^{-z}}{2}$; $\quad \cosh z = \dfrac{e^{z} + e^{-z}}{2}$; branches of $\log z$ wait for the contour-integration chapter

• Conformality theorem: $f$ analytic with $f'(z_0) \ne 0 \Rightarrow$ conformal at $z_0$ — preserves angle magnitude and sense
• Local picture: $\Delta w \approx f'(z_0)\,\Delta z$, with $f'(z_0) = \rho\,e^{i\alpha}$ — rotate by $\alpha = \arg f'(z_0)$, scale by $\rho = |f'(z_0)|$
• Critical point: a zero of $f'$. Angles are multiplied by $m$ (order of the first non-vanishing derivative); for $w = z^2$, $m = 2$, so angles double at $z = 0$
• Elementary maps: $w = az + b$ ($a \ne 0$) a similarity, conformal everywhere; $w = z^2$: $re^{i\theta} \mapsto r^2 e^{2i\theta}$; $w = 1/z$ conformal wherever defined (no critical points)
• Isogonal = angle magnitude preserved; conformal = isogonal + sense preserved; anti-conformal = isogonal + sense reversed (e.g. $w = \bar z$, not analytic)

• Möbius (bilinear) map: $w = \dfrac{az + b}{cz + d}$, $\quad ad - bc \ne 0$; conformal wherever defined since $w' = \dfrac{ad - bc}{(cz + d)^2} \ne 0$ (no critical points)
• Three elementary moves: translation $z + b$; rotation–magnification $az$ ($a = re^{i\theta}$ — rotate by $\theta$, scale by $r$); inversion $1/z$. Decomposition ($c \ne 0$): $w = \dfrac{a}{c} + \dfrac{bc - ad}{c\,(cz + d)}$
• Fixed points (set $w = z$) solve $cz^2 + (d - a)z - b = 0$ — two distinct, one repeated (parabolic), or a complex-conjugate pair
• Cross-ratio map through three points: $\dfrac{(w - w_1)(w_2 - w_3)}{(w - w_3)(w_2 - w_1)} = \dfrac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}$
• $\infty$ rule: delete the two factors containing the infinite point. Source $z_3 = \infty \Rightarrow$ right side $\dfrac{z - z_1}{z_2 - z_1}$; target $w_3 = \infty \Rightarrow$ left side $\dfrac{w - w_1}{w_2 - w_1}$
• Inverse: $z = \dfrac{dw - b}{-cw + a}$ — itself Möbius, same determinant
• Lines and circles share $A(x^2 + y^2) + Bx + Cy + D = 0$ (line if $A = 0$, circle if $A \ne 0$). Möbius maps send this family to itself; inversion is the only move that swaps a line for a circle, e.g. $\operatorname{Re}(z) = 1 \xrightarrow{\,1/z\,} \left|w - \tfrac12\right| = \tfrac12$

• $\displaystyle\int_C f(z)\,dz = \int_a^b f(z(t))\,z'(t)\,dt$; reversing orientation negates the integral
• Fundamental integral: $\displaystyle\oint_{|z - z_0| = r} (z - z_0)^n\,dz = 2\pi i$ if $n = -1$, else $0$ (any $r$, any $z_0$)
• Cauchy-Goursat: $f$ analytic on and inside a simple closed contour $C \Rightarrow \oint_C f(z)\,dz = 0$
• Path independence: $\int_C f\,dz = F(z_1) - F(z_0)$ when $F' = f$
• ML inequality: $\left|\int_C f\,dz\right| \le ML$, with $M = \max_C|f|$, $L = \operatorname{length}(C)$
• Deformation: deform $C$ through any region where $f$ is analytic without changing $\oint_C f\,dz$; the value depends only on the enclosed singularities
• $\operatorname{Log} z = \ln|z| + i\,\operatorname{Arg} z$, cut on $(-\infty, 0]$; $\frac{d}{dz}\operatorname{Log} z = \frac1z$ off the cut

• $f(z_0) = \dfrac{1}{2\pi i}\displaystyle\oint_C \dfrac{f(z)}{z - z_0}\,dz$
• Derivatives: $f^{(n)}(z_0) = \dfrac{n!}{2\pi i}\displaystyle\oint_C \dfrac{f(z)}{(z - z_0)^{n+1}}\,dz$
• Cauchy's inequality: $\left|f^{(n)}(z_0)\right| \le \dfrac{n!\,M_R}{R^{\,n}}$, with $M_R = \max_{|z - z_0| = R}|f(z)|$
• Liouville: a bounded entire function is constant
• Maximum modulus: a non-constant analytic $f$ attains $\max|f|$ only on the boundary
• Fundamental Theorem of Algebra: every nonconstant polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots, counting multiplicity

• Taylor (analytic on $|z - z_0| < R$): $f(z) = \displaystyle\sum_{n=0}^{\infty}\dfrac{f^{(n)}(z_0)}{n!}(z - z_0)^n$; radius $=$ distance to the nearest singularity
• Laurent (on an annulus, named by an inequality, e.g. valid in $1 < |z| < 2$): $f(z) = \displaystyle\sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$, with $a_{-1} = \operatorname{Res}_{z_0} f$
• Classify by the principal part: no negative powers $=$ removable; lowest term $(z - z_0)^{-m}$ $=$ pole of order $m$ ($m = 1$ simple); infinitely many $=$ essential (e.g. $e^{1/z}$)
• Simple pole: $\operatorname{Res}_{z_0} f = \lim_{z\to z_0}(z - z_0)f(z)$; for $f = p/q$ with a simple zero of $q$: $\operatorname{Res}_{z_0} f = \dfrac{p(z_0)}{q'(z_0)}$
• Pole of order $m$: $\operatorname{Res}_{z_0} f = \dfrac{1}{(m-1)!}\displaystyle\lim_{z\to z_0}\dfrac{d^{m-1}}{dz^{m-1}}\big[(z - z_0)^m f(z)\big]$
• Residue theorem: $\displaystyle\oint_C f\,dz = 2\pi i\sum \operatorname{Res} f$ over the poles enclosed by $C$
• Real trig integral: $\displaystyle\int_0^{2\pi} R(\cos\theta,\sin\theta)\,d\theta$ via $z = e^{i\theta}$, $\cos\theta = \tfrac12(z + z^{-1})$, $\sin\theta = \tfrac{1}{2i}(z - z^{-1})$, $d\theta = \dfrac{dz}{iz}$ — a unit-circle integral
• Rouché's theorem & the argument principle — recognition only in this course; full problem-solving not covered unless your class notes require it

• $\displaystyle\iint_R f\,dA = \lim \sum f\,\Delta A$ (pillar sum); equals the volume under $z = f(x,y)$ when $f \ge 0$; $\operatorname{Area}(R) = \displaystyle\iint_R 1\,dA$
• Vertical slices $\displaystyle\int_a^b\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx$; horizontal $\displaystyle\int_c^d\!\int_{h_1(y)}^{h_2(y)} f\,dx\,dy$ — outer limits constant, inner limits depend only on the outer variable
• Fubini: both orders agree for $f$ continuous on $R$; changing order $=$ re-sketch and re-describe the region, never relabel
• Polar: $x = r\cos\theta,\ y = r\sin\theta,\ x^2 + y^2 = r^2,\ dA = r\,dr\,d\theta$
• Mass $m = \displaystyle\iint_R \rho\,dA$; centroid $\bar{x} = \dfrac{1}{m}\iint_R x\rho\,dA,\ \ \bar{y} = \dfrac{1}{m}\iint_R y\rho\,dA$

• Jacobian $J = \dfrac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix}$; area element $dA = |J|\,du\,dv$ (absolute value — always)
• $\displaystyle\iint_R f(x,y)\,dx\,dy = \iint_S f\big(x(u,v), y(u,v)\big)\,|J|\,du\,dv$ — rewrite the integrand and the region in $u, v$ too
• Reciprocal shortcut: $\dfrac{\partial(x,y)}{\partial(u,v)} = \dfrac{1}{\partial(u,v)/\partial(x,y)}$ — compute the easier determinant, then reciprocate
• Polar is the special case: $x = r\cos\theta,\ y = r\sin\theta \Rightarrow |J| = r$, recovering $dA = r\,dr\,d\theta$
• Gaussian: $\left(\displaystyle\int_{-\infty}^{\infty} e^{-x^2}\,dx\right)^{2} = \iint_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dA = \pi \Rightarrow \displaystyle\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$

• Triple integral by shadow + slice: $\displaystyle\iiint_T f\,dV = \iint_{\text{shadow}}\!\left[\int_{\text{floor}}^{\text{roof}} f\,dz\right]dA$ — inner limits are surfaces, the shadow's limits are constant / outer-only
• Cylindrical $(r,\theta,z)$: $x=r\cos\theta,\ y=r\sin\theta$; $\ dV = r\,dr\,d\theta\,dz$. Use for axial symmetry / circular shadow / integrand in $x^2+y^2$
• Spherical $(\rho,\phi,\theta)$, $\phi$ from the $+z$-axis: $x=\rho\sin\phi\cos\theta,\ y=\rho\sin\phi\sin\theta,\ z=\rho\cos\phi$; $\ dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$. Use for spheres / cones / origin-radial symmetry
• The $r$ and $\rho^2\sin\phi$ are the Jacobians of these coordinate changes (Ch 11) — never optional
• Volume $=\displaystyle\iiint_T 1\,dV$; mass $=\displaystyle\iiint_T \delta\,dV$; centroid $\bar z = \dfrac{\iiint_T z\,\delta\,dV}{\text{mass}}$ (and likewise $\bar x,\bar y$)
• Useful: ball $\tfrac43\pi a^3$; $\int_0^\pi \sin\phi\,d\phi = 2$; uniform solid hemisphere centroid $\bar z = \tfrac{3a}{8}$

• Line integral / work: $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r} = \int_C P\,dx + Q\,dy$ — parametrize $\mathbf{r}(t)$, substitute, integrate in $t$
• Green's theorem (CCW, region $R$ on the left): $\displaystyle\oint_C P\,dx + Q\,dy = \iint_R\!\left(\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}\right)dA$; clockwise negates
• Area by the boundary (planimeter): $\displaystyle A = \tfrac12\oint_C (x\,dy - y\,dx)$ (CCW)
• Conservative test: $\dfrac{\partial P}{\partial y} = \dfrac{\partial Q}{\partial x}$ on a simply-connected region $\Rightarrow \mathbf{F} = \nabla f$, path-independent, every loop $= 0$ (the real-variable face of Ch 7's path independence)
• The Unit-1 bridge: $\displaystyle\oint_C f\,dz = \oint_C(u\,dx - v\,dy) + i\oint_C(v\,dx + u\,dy)$; Green's $+$ the CR equations make both integrands vanish $\Rightarrow \oint_C f\,dz = 0$ — Green's theorem $+$ CR $=$ Cauchy–Goursat
• A singularity enclosed by $C$ breaks Green's hypothesis: $\displaystyle\oint_{|z|=1}\tfrac1z\,dz = 2\pi i$ — the leftover is the residue

• Surface element: $dS = |\mathbf{r}_u \times \mathbf{r}_v|\,du\,dv$; vector element $d\mathbf{S} = \hat{\mathbf{n}}\,dS = \pm(\mathbf{r}_u \times \mathbf{r}_v)\,du\,dv$
• Graph $z=f(x,y)$: $dS = \sqrt{1+f_x^2+f_y^2}\,dA$; upward normal vector $(-f_x,-f_y,1)$
• Scalar surface integral: $\displaystyle\iint_S f\,dS = \iint_D f(\mathbf{r}(u,v))\,|\mathbf{r}_u\times\mathbf{r}_v|\,du\,dv$ (mass with $f=$ density; area with $f=1$)
• Flux: $\displaystyle\iint_S \mathbf{F}\cdot\hat{\mathbf{n}}\,dS = \iint_D \mathbf{F}\cdot(\mathbf{r}_u\times\mathbf{r}_v)\,du\,dv$; the sign follows the chosen orientation
• 3-D curl: $\operatorname{curl}\mathbf{F} = \nabla\times\mathbf{F} = (R_y-Q_z,\;P_z-R_x,\;Q_x-P_y)$
• Stokes: $\displaystyle\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\operatorname{curl}\mathbf{F})\cdot\hat{\mathbf{n}}\,dS$, $C$ oriented by the right-hand rule about $\hat{\mathbf{n}}$; the flat case $(\hat{\mathbf{n}}=\mathbf{k})$ is Green's theorem
• Surface independence: $\displaystyle\iint_S (\operatorname{curl}\mathbf{F})\cdot\hat{\mathbf{n}}\,dS$ depends only on the boundary $C$ — any surface with the same rim gives the same value

• Divergence: $\operatorname{div}\mathbf{F} = \nabla\cdot\mathbf{F} = P_x + Q_y + R_z$
• Divergence (Gauss) theorem: $\displaystyle\iint_S \mathbf{F}\cdot\hat{\mathbf{n}}\,dS = \iiint_V (\operatorname{div}\mathbf{F})\,dV$, $S$ the closed, outward-oriented boundary of the solid $V$
• The three bridges, one template — boundary integral $=$ interior integral of a derivative: $\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_D (Q_x-P_y)\,dA$ (Green) · $\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\operatorname{curl}\mathbf{F})\cdot\hat{\mathbf{n}}\,dS$ (Stokes) · $\iint_S \mathbf{F}\cdot\hat{\mathbf{n}}\,dS = \iiint_V (\operatorname{div}\mathbf{F})\,dV$ (Gauss)
• Same shape as Cauchy's integral formula (Unit 1): boundary values fix the interior

Pre-flight checklist (complete — run before writing any final answer)

• Region/solid sketched first? Limits depend only on the outer variables?
• Changing the order? Re-read the region from the sketch — don't just swap the limits.
• Jacobian present for a change of variables? ($r$ for polar/cylindrical, $\rho^2\sin\phi$ for spherical)
• Volume/area element right for the coordinate system?
• Flux: is the normal the one asked for (outward / upward / downward)? The sign follows it.
• Stokes/Green: does the boundary-curve direction match the normal by the right-hand rule?
• Closed surface → divergence-theorem candidate; open surface → integrate directly or close it off with a cap.
• Scalar surface integral uses $|\mathbf{r}_u\times\mathbf{r}_v|$ (magnitude); flux keeps the vector $\mathbf{r}_u\times\mathbf{r}_v$ (sign).