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If $f$ is contour-integrable along a path $g$, then $f$ is contour-integrable along the reverse path of $g$. |
If $g$ is a valid path, then $f$ is contour-integrable along $g$ if and only if $f$ is contour-integrable along the reverse path of $g$. |
If $g$ is a valid path, then the contour integral of $f$ along the reverse path of $g$ is the negative of the contour integral of $f$ along $g$. |
If $f$ has a contour integral along two paths $g_1$ and $g_2$, then $f$ has a contour integral along the path $g_1 + g_2$. |
If $f$ is contour integrable on two paths $g_1$ and $g_2$, and $g_1$ and $g_2$ are valid paths, then $f$ is contour integrable on the path $g_1 + g_2$. |
If $f$ is contour-integrable on the join of two paths $g_1$ and $g_2$, and $g_1$ is a valid path, then $f$ is contour-integrable on $g_1$. |
If $f$ is contour-integrable on the join of two paths $g_1$ and $g_2$, and $g_2$ is a valid path, then $f$ is contour-integrable on $g_2$. |
If $g_1$ and $g_2$ are valid paths, then $f$ is contour-integrable on $g_1 + g_2$ if and only if $f$ is contour-integrable on $g_1$ and $f$ is contour-integrable on $g_2$. |
If $f$ is integrable on two paths $g_1$ and $g_2$, and $g_1$ and $g_2$ are valid paths, then the integral of $f$ on the path $g_1 + g_2$ is equal to the sum of the integrals of $f$ on $g_1$ and $g_2$. |
If $f$ has a contour integral along a path $g$, then $f$ has a contour integral along the path $g$ shifted by $a$. |
If $f$ has a contour integral along a path $g$ shifted by $a$, then $f$ has a contour integral along $g$. |
If $g$ is a closed path, then $f$ has a contour integral along $g$ if and only if $f$ has a contour integral along the path $g$ shifted by $a$. |
If $g$ is a closed path, then $f$ is integrable on $g$ if and only if $f$ is integrable on $g$ shifted by $a$. |
If $g$ is a closed path, then the contour integral of $g$ is the same as the contour integral of $g$ shifted by $a$. |
The contour integral of $f$ along the line segment from $a$ to $b$ is equal to the integral of $f(a + t(b - a))(b - a)$ from $0$ to $1$. |
The contour integral of a function $f$ along the path $a \mapsto a$ is zero. |
The contour integral of a function $f$ along the path $a \mapsto a$ is $0$ if and only if $i=0$. |
The integral of a function $f$ along a path that is a single point is zero. |
If $f$ is a function that is integrable over a path $g$, then $f$ is integrable over the subpath $g$ from $u$ to $u$. |
If $f$ is a contour-integrable function on a path $g$, then $f$ is contour-integrable on the subpath of $g$ from $u$ to $u$. |
The contour integral of a function $f$ along a subpath of a path $g$ from $u$ to $u$ is zero. |
If $f$ is integrable on a path $g$, then $f$ is integrable on any subpath of $g$. |
If $f$ is contour-integrable on a path $g$, then $f$ is contour-integrable on any subpath of $g$. |
If $f$ is integrable on a path $g$, then $f$ is integrable on any subpath of $g$. |
If $f$ is integrable on a path $g$, then the integral of $f$ on the subpath of $g$ from $u$ to $v$ is equal to the integral of $f(g(x)) \cdot g'(x)$ on the interval $[u,v]$. |
If $f$ is integrable on a path $g$, and $u,v,w$ are points on $g$ with $u<v<w$, then the integral of $f$ on the subpath from $u$ to $v$ plus the integral of $f$ on the subpath from $v$ to $w$ is equal to the integral of $f$ on the subpath from $u$ to $w$. |
If $f$ is integrable on a path $g$, then the integral of $f$ on the subpath of $g$ from $u$ to $v$ plus the integral of $f$ on the subpath of $g$ from $v$ to $w$ is equal to the integral of $f$ on the subpath of $g$ from $u$ to $w$. |
The contour integral of a function $f$ along a curve $g$ is equal to the integral of $f \circ g$ times the derivative of $g$. |
If two paths are equal and the integrands are equal on the path images, then the contour integrals are equal. |
If $f$ is a complex-valued function defined on the real line, then the contour integral of $f$ along the line segment from $a$ to $b$ is equal to the integral of $f$ along the line segment from $a$ to $b$. |
If $f$ is a complex-valued function defined on the real line, then $f$ is integrable on the interval $[a,b]$ if and only if $f$ is contour-integrable on the line segment from $a$ to $b$. |
If $f$ is a complex-valued function defined on the real line, then the contour integral of $f$ along the line segment from $a$ to $b$ is equal to the integral of $f$ along the line segment from $a$ to $b$. |
If $f$ is differentiable on $S$ and $g$ is piecewise differentiable on $[a,b]$ with $g([a,b]) \subseteq S$, then $\int_a^b f'(g(x)) g'(x) dx = f(g(b)) - f(g(a))$. |
If $f$ is differentiable on a set $S$ and $g$ is a path in $S$, then $\int_g f' = f(b) - f(a)$, where $a$ and $b$ are the endpoints of $g$. |
If $f$ is differentiable on a set $S$ and $g$ is a closed path in $S$, then $\int_g f'(z) dz = 0$. |
If $f$ is continuous on the closed segment $[a,b]$, then $f$ is integrable on the line segment $[a,b]$. |
The function $f(x) = x^2/2$ has derivative $f'(x) = x$. |
The integral of the identity function over a line segment is the square of the length of the segment divided by two. |
The constant function $f(x) = c$ is integrable on the line segment from $a$ to $b$. |
The function $f(x) = x$ is integrable on the line segment from $a$ to $b$. |
If $f$ has a contour integral $i$ along a contour $g$, then $-f$ has a contour integral $-i$ along $g$. |
If $f_1$ and $f_2$ have contour integrals $i_1$ and $i_2$ along a contour $g$, then $f_1 + f_2$ has contour integral $i_1 + i_2$ along $g$. |
If $f_1$ and $f_2$ have contour integrals $i_1$ and $i_2$ along a contour $g$, then $f_1 - f_2$ has contour integral $i_1 - i_2$ along $g$. |
If $f$ has a contour integral $i$ along $g$, then $cf$ has a contour integral $ci$ along $g$. |
If $f$ has a contour integral $i$ along $g$, then $cf$ has a contour integral $ci$ along $g$. |
If $f$ has a contour integral $i$ along $g$, then $f/c$ has a contour integral $i/c$ along $g$. |
If $f$ and $g$ are functions that agree on the image of a path $p$, and if $f$ has a contour integral along $p$, then $g$ has the same contour integral along $p$. |
If $f$ has a contour integral along the line segment from $a$ to $b$, and $f$ is bounded by $B$ on the line segment, then the absolute value of the contour integral is bounded by $B$ times the length of the line segment. |
The contour integral of a constant function along a line segment is the product of the constant and the length of the line segment. |
The contour integral of the zero function is zero. |
If $f$ is identically zero on the image of a path $g$, then the contour integral of $f$ along $g$ is zero. |
If $f_1, \ldots, f_n$ are functions that have contour integrals $i_1, \ldots, i_n$ along a path $p$, then the function $f_1 + \cdots + f_n$ has contour integral $i_1 + \cdots + i_n$ along $p$. |
The integral of a constant function along a line segment is equal to the product of the constant and the length of the line segment. |
If $f$ is integrable on a contour $g$, then $-f$ is integrable on $g$ and $\int_g -f = -\int_g f$. |
If $f_1$ and $f_2$ are integrable on a contour $g$, then $f_1 + f_2$ is integrable on $g$ and $\int_g (f_1 + f_2) = \int_g f_1 + \int_g f_2$. |
If $f_1$ and $f_2$ are integrable on a contour $g$, then $f_1 - f_2$ is integrable on $g$ and $\int_g (f_1 - f_2) = \int_g f_1 - \int_g f_2$. |
If $f$ is integrable on a contour $g$, then $c f$ is integrable on $g$ and $\int_g (c f) = c \int_g f$. |
If $f$ is integrable on a contour $g$, then $f(x)c$ is integrable on $g$ and $\int_g f(x)c = \int_g f(x)c$. |
If $f$ is integrable on a contour $g$, then $\int_g f(x) \, dx = \int_g \frac{f(x)}{c} \, dx$. |
If two functions $f$ and $g$ are equal on the image of a path $p$, then the contour integral of $f$ along $p$ is equal to the contour integral of $g$ along $p$. |
If $f$ is a function that is identically zero on the image of a path $g$, then the contour integral of $f$ along $g$ is zero. |
If $f$ is integrable on the line segment from $a$ to $b$, and $f$ is bounded by $B$ on the line segment, then the integral of $f$ on the line segment is bounded by $B$ times the length of the line segment. |
The contour integral of the zero function is zero. |
If $f_1, \ldots, f_n$ are contour integrable functions on a contour $p$, then the function $f(x) = f_1(x) + \cdots + f_n(x)$ is also contour integrable on $p$, and $\int_p f = \int_p f_1 + \cdots + \int_p f_n$. |
If $f$ is contour-integrable on a path $p$ and $f(x) = g(x)$ for all $x$ in the image of $p$, then $g$ is contour-integrable on $p$. |
If $f$ is contour-integrable on a contour $g$, then $-f$ is contour-integrable on $g$. |
If $f_1$ and $f_2$ are contour integrable on $g$, then $f_1 + f_2$ is contour integrable on $g$. |
If $f_1$ and $f_2$ are contour integrable on $g$, then $f_1 - f_2$ is contour integrable on $g$. |
If $f$ is contour-integrable on a contour $g$, then $cf$ is contour-integrable on $g$. |
If $f$ is contour-integrable on $g$, then $f \cdot c$ is contour-integrable on $g$. |
If $f$ is contour-integrable on a contour $g$, then $f/c$ is contour-integrable on $g$ for any constant $c$. |
If $f_1, \ldots, f_n$ are contour-integrable functions, then so is $f_1 + \cdots + f_n$. |
If $f$ has a contour integral along the line segment from $a$ to $b$, then $f$ has a contour integral along the line segment from $b$ to $a$, and the value of the contour integral is the negative of the value of the contour integral along the line segment from $a$ to $b$. |
If $f$ is continuous on the closed segment from $a$ to $b$, then the contour integral of $f$ along the line segment from $a$ to $b$ is equal to the negative of the contour integral of $f$ along the line segment from $b$ to $a$. |
If $f$ has a contour integral along the line segment from $a$ to $c$ and along the line segment from $c$ to $b$, then $f$ has a contour integral along the line segment from $a$ to $b$. |
If $f$ is continuous on the closed segment $[a,b]$, then $f$ is continuous on the closed segment $[a,c]$, where $c$ is a point on the segment $[a,b]$. |
If $f$ is a continuous function on the closed line segment from $a$ to $b$, then the contour integral of $f$ along the line segment from $a$ to $b$ is equal to the sum of the contour integrals of $f$ along the line segments from $a$ to $c$ and from $c$ to $b$. |
If $f$ is continuous on the closed segment $[a,b]$, then the contour integral of $f$ along the line segment $[a,b]$ is equal to the sum of the contour integrals of $f$ along the line segments $[a,c]$ and $[c,b]$. |
If the interval $[c,d]$ is non-empty, then the image of the interval $[(a,c),(b,d)]$ under the function $fst$ is the interval $[a,b]$. |
If $a < b$, then the image of the box $[a,b] \times [c,d]$ under the projection onto the second coordinate is the interval $[c,d]$. |
If $f$ is continuous on the image of the paths $g$ and $h$, and $g$ and $h$ are valid paths, then the contour integral of $g$ around $h$ is equal to the contour integral of $h$ around $g$. |
If $\gamma$ is a valid path, then $-\gamma$ is a valid path. |
If $f$ has a contour integral along $\gamma$, then $f$ has a contour integral along $-\gamma$. |
If $f$ is contour-integrable on $\gamma$, then $f$ is contour-integrable on $-\gamma$. |
If $p$ is a polynomial function, then $p$ is continuously differentiable on $S$. |
If $p$ is a polynomial function, then $p$ is a valid path. |
If $z \neq g(x)$, then the subpath of $g$ from $x$ to $x$ is a valid path. |
The starting point of a part of a circle is the center of the circle plus the radius times the complex exponential of the starting angle. |
The endpoint of the arc of a circle with radius $r$ and center $z$ from angle $s$ to angle $t$ is $z + r\exp(it)$. |
The reverse of a part of a circle path is the same part of the circle path, but with the start and end points swapped. |
The derivative of the function $f(x) = \exp(i \cdot x)$ is $f'(x) = i \cdot \exp(i \cdot x)$. |
The function $f(t) = c + r e^{it}$ is differentiable at $x$ on the set $A$. |
The derivative of the part of a circle path from $s$ to $t$ is $\iota r (t - s) \exp(\iota \text{ linepath } s t x)$. |
If $0 \leq x \leq 1$, then the derivative of the function $f(x) = \exp(\iota \cdot \text{linepath}(s, t, x))$ at $x$ is $\iota \cdot r \cdot (t - s) \cdot \exp(\iota \cdot \text{linepath}(s, t, x))$. |
The path $t \mapsto z + r e^{it}$ is valid. |
The part of a circle is a path. |
The image of the path $z + r e^{i \theta}$ for $\theta \in [s, t]$ is the set of points $z + r e^{i \theta}$ for $\theta \in [s, t]$. |
The image of the path $z + r \exp(i \theta)$ for $\theta$ in the closed segment $[s, t]$ is the set of points $z + r \exp(i \theta)$ for $\theta$ in the closed segment $[s, t]$. |
If $s \leq t$ and $r \geq 0$, then the image of the path $t \mapsto z + r e^{it}$ for $s \leq t \leq t$ is contained in the sphere of radius $r$ centered at $z$. |
If $w$ is on the image of the path $z + r e^{i \theta}$ for $\theta \in [s, t]$, then $|w - z| = r$. |
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