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For steady incompressible flow and irrotational flow of constant mass density, with no body force, this formula gives the hydrodynamic torque about the origin per unit span on an impermeable body. Orient its boundary counterclockwise, and let positive torque be counterclockwise. The proof uses $M=\mathrm{Re}{\oint }_{C}pzd\bar{z}$, the Bernoulli equation $p=P_{\ast }-\rho |w^{\prime }{|}^2/2$, and the tangency identity $|w^{\prime }{|}^{2}d\bar{z}=(w^{\prime }{)}^{2}dz$. The real part of the constant-pressure term vanishes because $\mathrm{Re}(zd\bar{z})=d|z{|}^2/2$. The full complex integral of $zd\bar{z}$ need not vanish.

Place a held point vortex of circulation $\Gamma$ at real position $b>a$ outside a circular cylinder $|z|=a$, and prescribe circulation $-\Gamma$ about the cylinder. An opposite image at $c=a^2/b$ gives complex velocity $q(z)=-i\Gamma (1/(z-b)-1/(z-c))/(2\pi )$. On the positively oriented cylinder contour, the only pole of $q^2$ inside is at $c$, with residue $-{\Gamma }^2/(2{\pi }^2(b-c))$. The residue theorem and Blasius force theorem give $F_x-i{F}_y=\rho {\Gamma }^2/(2\pi (b-c))$. Thus $F_y=0$ and $F_x=\rho {\Gamma }^{2}b/(2\pi (b^2-a^2))>0$: the force on the cylinder points toward the vortex. The specified cylinder circulation is part of the hypothesis; a zero-circulation cylinder has an additional central image term.

For a two-dimensional steady ideal flow outside a body, let $q=w^{\prime }$ be the complex velocity. Suppose $q$ is holomorphic in the exterior and $q^2$ has a Laurent series near infinity, with coefficient $a_{-1}$ of $z^{-1}$. Orient the body contour counterclockwise and assume the usual impermeability and Bernoulli hypotheses of the Blasius force theorem. Deforming the contour to a large circle gives $\oint q^{2}dz=2\pi i{a}_{-1}$. Substitution in Blasius' formula proves $F_x-i{F}_y=-\pi \rho a_{-1}$.

In particular, $q(z)=q_{\infty }+O(z^{-2})$ makes the force zero: its square has no reciprocal term. This criterion handles branch-cut obstacles by applying the contour in their analytic exterior, rather than separately assigning forces to endpoint singularities.

The Milne-Thomson circle theorem places an image dipole inside a cylinder at the inverse point of an external dipole. In the residue of the squared complex velocity, only the cross term between the analytic external velocity and the image double pole contributes. The product of their conjugate dipole strengths is independent of orientation, so the cylinder force is radial toward the external dipole.

If holding one unit of a commodity costs a constant cash amount $\delta dt$ for storage, its gain is $dS-\delta dt$. A delta hedge long a derivative and short $V_S$ commodity units has riskless drift $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}+\delta V_S$ and value $V-S{V}_S$. Equating the drift to the bank return gives the displayed Black-Scholes equation. This is an additive cash carry term, unlike a proportional dividend or storage yield.

The corresponding pricing diffusion is $d\hat{S}=(r\hat{S}+\delta )dt+\sigma \hat{S}dW$. A fixed-strike forward has value $S+\delta (1-e^{-r\tau })/r-K{e}^{-r\tau }$, and put-call parity gives that same value for a matching call minus put. Replication buys the commodity and a bank reserve equal to the present value of its future storage payments. For $r=0$, the reserve is $\delta \tau$.

Differentiating the displayed transformation shows that it preserves the Black-Scholes equation. The first derivative coefficient fixes the exponent $1-2r/{\sigma }^2$, which also makes the zero-order coefficient correct. Subtracting this reflected image of a European call option from the call itself enforces a zero boundary at $S=B$. For strike price $K\ge B$, the reflected maturity contingent claim payoff vanishes on $S>B$, giving the no-rebate down-and-out option formula. Prices using this reflection are conditional on the barrier not having been hit already.

The bisection method preserves some root in a sign-changing bracket but does not select a fixed root when several are present. For the displayed polynomial, endpoint and midpoint roots are returned immediately. If $2<b<3$, the first midpoint lies in $(1/2,1)$ with negative function value, so the retained bracket contains only zero. If $b=3$, the first midpoint is exactly one. If $3<b\le 4$, that midpoint lies in $(1,3/2]$ with positive function value, so the retained bracket contains only two. At $b=2$ the right endpoint itself is the root two.

Thus arbitrarily small changes of the initial endpoint across $b=3$ change the limiting root from zero to two, with one selected exactly at the transition. The guarantee concerns distance to a root inside the retained interval, rather than continuous dependence of that root's identity on the starting data.

Two expressions are equal when they denote the same mathematical object. An equality test in a program implements a specified comparison of represented values; for example, approximate floating-point arithmetic comparison is not identical to exact equality of real numbers.

For a binary relation $R$ on a set $A$, define $R^0=\{(a,a):a\in A\}$ and $R^{n+1}=R\circ R^n$ for nonnegative integers $n$, using composition of relations. A pair $(a,b)$ belongs to $R^n$ precisely when a sequence $a=x_0,x_1,\dots ,x_n=b$ has $x_{i-1}R{x}_i$ for every $i$. This follows by mathematical induction, splitting off one step in the recurrence. Thus $R^n$ records paths with exactly $n$ steps, rather than at most $n$ steps. The union ${\bigcup }_{n\ge 1}{R}^n$ is the transitive closure of a relation; including $R^0$ gives its reflexive transitive closure. The powers of a single relation agree under the two common composition-order conventions.

Reversing every ordered pair of a binary relation. Swapping the two components is its own inverse and therefore preserves distinctness of a representation by a list of distinct pairs.

A binary relation is right Euclidean when any two successors of the same element are related in the indicated order. A reflexive relation with this property is an equivalence relation: reflexivity gives symmetry, and symmetry followed by the Euclidean property gives transitivity. Without reflexivity, even the empty binary relation is Euclidean.

A weighted quadratic Bézier curve with positive middle weight $w$ retains endpoint interpolation and endpoint tangent directions while changing the bulge. For a circular arc of angle $\theta$ less than $180^{\circ }$, choose endpoints on the circle, their tangent intersection as the middle control point, and $w=\cos (\theta /2)$. Unlike an ordinary nonconstant polynomial Bézier curve, this rational representation can describe a circular arc exactly.

Use the convention $K_m={\sum }_{|j|\le m}(1-|j|/(m+1))e^{ijt}$. Multiplying $K_{n-1}$ by $2n\sin (nt)$ gives Fourier coefficients $-ik$ for $|k|\le n$. A trigonometric polynomial of degree at most $n$ has no other frequencies, so convolution proves the displayed identity. Positivity and integral one of the Fejér kernel then give $\Vert P^{\prime }{\Vert }_{\infty }\le 2n\Vert P{\Vert }_{\infty }$. This is a useful elementary estimate; the sharp Bernstein inequality for trigonometric polynomials improves $2n$ to $n$.

In the area Hilbert space of the unit disk, conjugates of Bergman space functions vanishing at zero are orthogonal to all holomorphic functions, since ${\int }_{D}hgdA=\pi h(0)g(0)$. This follows from the angular mean-value property and the Cauchy-Schwarz inequality, which makes the product integrable. But the radial function $|z{|}^2-1/2$ is also orthogonal to every holomorphic function: its radial integral against the constant angular mean is zero. It is not conjugate holomorphic, since it is real-valued and nonconstant. Thus the conjugate-holomorphic family is a proper subset of the orthogonal complement.

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

A beetle of the family Meloidae, several members of which supply cantharidin.