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If $g$ is holomorphic on an open set $S$ and $g(z) = (z - a)f(z)$ for all $z \in S - \{a\}$, then $f$ is holomorphic on $S$. |
Suppose $g$ is holomorphic on a set $S$ and $a$ is an interior point of $S$. Suppose that for all $z \in S - \{a\}$, we have $g(z) = (z - a)f(z)$. Suppose that $f(a) = g'(a)$ and $g(a) = 0$. Then $f$ is holomorphic on $S$. |
If $g$ is holomorphic on an open set $S$ and $g(z) = (z - a)f(z)$ for all $z \in S - \{a\}$, then $f$ is holomorphic on $S$. |
If $g$ is analytic on $S$ and for every $z \in S$, there exists a $d > 0$ such that for all $w \in B(z, d) - \{a\}$, we have $g(w) = (w - a)f(w)$, then $F$ is analytic on $S$, where $F(z) = f(z) - g(a)/(z - a)$ if $z \neq a$ and $F(a) = g'(a)$. |
If $g$ is analytic on $S$ and for all $z \in S$, there exists a $d > 0$ such that for all $w \in B(z, d) - \{a\}$, $g(w) = (w - a)f(w)$, then $f$ is analytic on $S$. |
If $g$ is analytic on $S$ and $S \subseteq T$ where $T$ is open, and if $g$ has a pole at $a$ with residue $g(a)$, then $f(z) = \frac{g(z)}{z - a}$ is analytic on $S$. |
If $g$ is an analytic function on a set $S$ and $S$ is a subset of an open set $T$ such that $g$ has a pole of order $1$ at $a$ and $g(a) = 0$, then $f$ is analytic on $S$. |
If $F$ is a continuous function from $U \times U$ to $\mathbb{C}$ and $F(w)$ is contour integrable for all $w \in U$, then the function $w \mapsto \int_{a}^{b} F(w)$ is continuous on $U$. |
If $f$ is holomorphic on an open set $U$, and $\gamma$ is a polynomial function such that the image of $\gamma$ is contained in $U - \{z\}$, then the contour integral of $f(w)/(w-z)$ along $\gamma$ is equal to $2\pi i$ times the winding number of $\gamma$ around $z$ times $f(z)$. |
If $f$ is holomorphic on an open set $S$, and $\gamma$ is a closed path in $S$ that does not pass through $z$, then the integral of $f(w)/(w-z)$ along $\gamma$ is $2\pi i$ times the winding number of $\gamma$ around $z$ times $f(z)$. |
If $f$ is holomorphic on an open set $S$, and $\gamma$ is a closed path in $S$, then $\int_\gamma f(z) dz = 0$. |
If $f$ is holomorphic on an open set $S$ and $\gamma$ is a closed path in $S$, then $\int_\gamma f(z) dz = 0$. |
If $S$ is a simply connected set and $g$ is a closed path in $S$ that does not pass through $z$, then the winding number of $g$ about $z$ is zero. |
If $S$ is an open simply connected set, $f$ is a holomorphic function on $S$, and $g$ is a closed path in $S$, then $\int_g f(z) dz = 0$. |
If $f$ is a holomorphic function on a convex open set $A$ such that $f(x) \neq 0$ for all $x \in A$, then there exists a holomorphic function $g$ on $A$ such that $e^{g(x)} = f(x)$ for all $x \in A$. |
If $f$ is holomorphic on a ball $B(z,r)$ and continuous on the closed ball $\overline{B}(z,r)$, and if $f(B(z,r)) \subset B(y,B_0)$, then the $n$th derivative of $f$ at $z$ is bounded by $(n!)B_0/r^n$. |
If $f$ is holomorphic on a ball $B(\xi, r)$ and continuous on the closed ball $\overline{B}(\xi, r)$, and if $f$ is bounded on the boundary of the ball, then the $n$th derivative of $f$ at $\xi$ is bounded by a constant times $B/r^n$. |
If $f$ is a holomorphic function on $\mathbb{C}$ and there exist constants $A$ and $B$ such that $|f(z)| \leq B|z|^n$ for all $z$ with $|z| \geq A$, then $f$ is a polynomial of degree at most $n$. |
If $f$ is a bounded holomorphic function on the complex plane, then $f$ is constant. |
If $f$ is a power series with radius of convergence $r > 0$ and $f(\xi) = 0$, then there exists $s > 0$ such that $f(z) \neq 0$ for all $z \<in> \mathbb{C}$ with $|z - \xi| < s$ and $z \neq \xi$. |
Suppose $f$ is holomorphic on the ball $B(z_0, r)$. Then the radius of convergence of the power series expansion of $f$ at $z_0$ is at least $r$. Moreover, the power series expansion of $f$ at $z_0$ converges to $f$ on the ball $B(z_0, r)$. |
If $f$ is a complex power series such that $f(z) \neq 0$ for all $z$ with $|z| < r$, then the radius of convergence of $f^{-1}$ is at least $\min(r, \text{radius of convergence of } f)$. Moreover, if $|z| < \text{radius of convergence of } f$ and $|z| < r$, then $f^{-1}(z) = 1/f(z)$. |
If $f$ and $g$ are two complex power series with positive convergence radii, and $g$ is nonzero on the ball of radius $r$, then the convergence radius of $f/g$ is at least $r$. |
If $f$ and $g$ are power series with $g$ having a lower degree than $f$, and $g$ has a nonzero constant term, then the power series $f/g$ has a radius of convergence at least as large as the minimum of the radii of convergence of $f$ and $g$. |
If $f$ is holomorphic on an open set $A$ containing $0$, then $f$ has a power series expansion about $0$. |
The radius of convergence of the Taylor series of $\tan(z)$ at $c$ is at least $\pi / (2 \|c\|)$. |
If $|z| < \pi / (2 |c|)$, then the Taylor series for $\tan(cz)$ at $0$ converges to $\tan(cz)$. |
If $f$ is a function from a convex set $U$ to the space of functions from $[a, b]$ to $\mathbb{C}$ such that $f$ is differentiable in $x$ and $f(x)$ is integrable for all $x \in U$, then the function $g(x) = \int_a^b f(x)(t) dt$ is holomorphic on $U$. |
The natural logarithm function is measurable. |
If $f$ and $g$ are measurable functions from a measurable space $M$ to the real numbers, then the function $x \mapsto f(x)^{g(x)}$ is measurable. |
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 $a$ to the midpoint of $a$ and $b$, and $f$ has a contour integral along the line segment from the midpoint of $a$ and $b$ to $b$, then $f$ has a contour integral along the line segment from... |
If $f$ is continuous 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 the midpoint of $a$ and $b$, and from the midpoint of $a$ and $b$ to $b$. |
If $m$ is a real number and $a, b, c$ are points in the plane, then the integral of the function $f(x) = mx + d$ over the triangle with vertices $a, b, c$ is zero. |
If $f$ has a contour integral along the path $a \rightarrow b \rightarrow c \rightarrow d$, then the contour integral of $f$ along the path $a \rightarrow b$ plus the contour integral of $f$ along the path $b \rightarrow c$ plus the contour integral of $f$ along the path $c \rightarrow d$ is equal to the contour integr... |
If $f$ has a contour integral along the path $a \rightarrow b \rightarrow c \rightarrow d \rightarrow e$, then the sum of the contour integrals of $f$ along the paths $a \rightarrow b$, $b \rightarrow c$, $c \rightarrow d$, and $d \rightarrow e$ is equal to the contour integral of $f$ along the path $a \rightarrow b \r... |
If the norm of the sum of two numbers is at least $e$, then the norm of one of the numbers is at least $e/2$. |
If $e \leq \|a + b + c + d\|$, then at least one of $\|a\|$, $\|b\|$, $\|c\|$, or $\|d\|$ is at least $e/4$. |
Suppose $f$ is a continuous function defined on the convex hull of three points $a$, $b$, and $c$. If the distances between these points are bounded by $K$, and if the integral of $f$ around the triangle formed by these points is at least $eK^2$, then there exists a triangle with vertices $a'$, $b'$, and $c'$ such that... |
If $x$ and $y$ are points in the convex hull of $a$, $b$, and $c$, then the distance between $x$ and $y$ is less than or equal to the distance between any two of $a$, $b$, and $c$. |
If $f$ is a continuous function on a set $S$ and $f$ is differentiable at a point $x \in S$, then there exists a positive number $k$ such that for any triangle with vertices $a$, $b$, and $c$ and with $x$ in the interior of the triangle, if the triangle is contained in $S$ and the sides of the triangle are less than $k... |
$f$ is at $n$. |
If $f(n+1)$ follows $f(n)$, then $f(n+2)$ follows $f(n+1)$, and so on. |
Suppose that we have a predicate $At(x,y,z,n)$ that holds for some $x_0, y_0, z_0$ and $n=0$, and that for any $x,y,z,n$ such that $At(x,y,z,n)$ holds, there exists $x',y',z'$ such that $At(x',y',z',n+1)$ holds and $Follows(x',y',z',x,y,z)$ holds. Then there exist functions $f,g,h$ such that $f(0) = x_0$, $g(0) = y_0$,... |
If $f$ is holomorphic on the triangle with vertices $a$, $b$, and $c$, then $\int_a^b f(z) dz + \int_b^c f(z) dz + \int_c^a f(z) dz = 0$. |
If $f$ is a continuous function on the convex hull of three points $a$, $b$, and $c$, and $c$ is a convex combination of $a$ and $b$, then the sum of the contour integrals of $f$ along the three sides of the triangle is zero. |
If $f$ is a continuous function on the triangle with vertices $a$, $b$, and $c$, and $c$ is a multiple of $b - a$, then the sum of the integrals of $f$ around the three sides of the triangle is zero. |
If $f$ is a continuous function defined on the triangle with vertices $a$, $b$, and $c$, and $f$ is holomorphic on the interior of the triangle, then $\int_\gamma f(z) dz = 0$, where $\gamma$ is the boundary of the triangle. |
If $f$ is continuous on the triangle with vertices $a$, $b$, and $c$, and if $f$ is differentiable at all points of the interior of the triangle except for a finite set $S$, then $\int_\gamma f(z) dz = 0$, where $\gamma$ is the triangle with vertices $a$, $b$, and $c$. |
If $a \in S$, $b,c \in S$, and $S$ is starlike with respect to $a$, then the convex hull of $a,b,c$ is contained in $S$. |
If $f$ is a continuous function on an open set $S$ and $a \in S$, and if the integral of $f$ around any triangle with vertices in $S$ is zero, then $f$ is a primitive of $f$ on $S$. |
Suppose $f$ is a continuous function defined on a starlike set $S$ and $f$ is complex differentiable on $S$ except for a finite number of points. Then there exists a function $g$ such that $g'(x) = f(x)$ for all $x \<in> S$. |
If $S$ is an open, starlike set, $f$ is a continuous function on $S$ that is differentiable on $S$ except for a finite set $k$, $g$ is a closed path in $S$, and $f$ is holomorphic on $S - k$, then $\int_g f(z) dz = 0$. |
If $f$ is holomorphic on an open, starlike set $S$, and $g$ is a closed path in $S$, then $\int_g f(z) dz = 0$. |
If $f$ is a continuous function on a convex set $S$ and $a \in S$, then the function $g(x) = \int_a^x f(t) dt$ is differentiable at every point $x \in S$ and $g'(x) = f(x)$. |
If $f$ is a continuous function on a convex set $S$ and $f$ has a contour integral of $0$ around every triangle in $S$, then $f$ is the derivative of some function $g$ on $S$. |
Suppose $f$ is a continuous function defined on a convex set $S$ and $f$ is differentiable at every point of $S$ except for a finite set $K$. Then there exists a function $g$ such that $g$ is differentiable at every point of $S$ and $g'(x) = f(x)$ for all $x \in S$. |
If $f$ is a holomorphic function on a convex open set $S$, then there exists a function $g$ such that $g'(x) = f(x)$ for all $x \in S$. |
If $f$ is a continuous function on a convex set $S$, and $f$ is differentiable on the interior of $S$ except for a finite number of points, then the integral of $f$ along any closed path in $S$ is zero. |
If $f$ is holomorphic on a convex set $S$ and $g$ is a closed path in $S$, then $\int_g f(z) dz = 0$. |
If $f$ is a continuous function on the closed disc of radius $e$ centered at $a$, and $f$ is differentiable on the open disc of radius $e$ centered at $a$ except at finitely many points, then the integral of $f$ around the boundary of the closed disc is zero. |
If $f$ is holomorphic on a disc, and $g$ is a closed path in the disc, then $\int_g f(z) dz = 0$. |
If $g$ is piecewise differentiable on $[a,b]$ and $f$ is differentiable on $g([a,b])$, then the function $x \mapsto f'(g(x)) \cdot g'(x)$ is integrable on $[a,b]$. |
If $g$ is piecewise differentiable on $[a,b]$ and $f$ is differentiable at every point in the image of $g$, then the composition $f \circ g$ is integrable on $[a,b]$. |
If $f$ is locally integrable on a set $S$ and $g$ is a path in $S$, then $f$ is integrable along $g$. |
If $f$ is a continuous function on an open set $S$ and $f$ is holomorphic on $S$ except for a finite number of points, then $f$ is contour integrable on any path in $S$. |
If $f$ is holomorphic on an open set $S$ and $g$ is a path in $S$, then $f$ is contour integrable along $g$. |
If $z$ is not in $S$, then the function $f(w) = 1/(w-z)$ is continuous on $S$. |
If $g$ is a valid path and $z$ is not in the image of $g$, then the function $1/(w-z)$ is contour-integrable along $g$. |
If $p$ is a path in an open set $S$, then there is a $\delta > 0$ such that if $g$ and $h$ are paths with $g(0) = h(0)$, $g(1) = h(1)$, and $g(t)$ and $h(t)$ are within $\delta$ of $p(t)$ for all $t \in [0,1]$, then $g$ and $h$ are also in $S$, and the contour integrals of $g$ and $h$ are equal for any holomorphic func... |
If $p$ is a path in an open set $S$, then there is a positive number $d$ such that if $g$ and $h$ are paths with $g(0) = h(0)$ and $g(1) = h(1)$ and $g(t)$ and $h(t)$ are within $d$ of $p(t)$ for all $t$, then $g$ and $h$ are paths in $S$ and $\int_g f = \int_h f$ for all holomorphic functions $f$ on $S$. Similarly, if... |
If $S$ is an open set and $g$ is a valid path with image contained in $S$, then there exists a constant $L$ such that for any holomorphic function $f$ on $S$ with $|f(z)| \leq B$ for all $z \in S$, we have $|\int_g f(z) dz| \leq LB$. |
If $g$ and $h$ are homotopic paths or loops in an open set $S$, then $\int_g f = \int_h f$ for any holomorphic function $f$ on $S$. |
If two paths are homotopic in an open set, then the contour integrals of a holomorphic function along the paths are equal. |
If two paths are homotopic, then their contour integrals are equal. |
If $f$ has a contour integral along a path $h$, and $f$ is contour integrable along a path $g$, and the contour integrals of $f$ along $g$ and $h$ are equal, then $f$ has a contour integral along $g$. |
If $f$ is holomorphic on an open set $S$, $g$ is a valid path in $S$, and $g$ is homotopic to a constant path, then $f$ has contour integral $0$ along $g$. |
The complex number $z$ is equal to the complex number with real part $Re(z)$ and imaginary part $Im(z)$. |
If the real and imaginary parts of two complex numbers are equal, then the complex numbers are equal. |
Two complex numbers are equal if and only if their real and imaginary parts are equal. |
The real part of the quotient of two complex numbers is equal to the quotient of the real parts of the two complex numbers. |
The imaginary part of the quotient of two complex numbers is given by the formula $\frac{\text{Im}(x) \cdot \text{Re}(y) - \text{Re}(x) \cdot \text{Im}(y)}{\text{Re}(y)^2 + \text{Im}(y)^2}$. |
The quotient of two complex numbers is given by the following formula. |
The real part of $x^2$ is equal to $(\text{Re}(x))^2 - (\text{Im}(x))^2$. |
The imaginary part of $x^2$ is twice the product of the real and imaginary parts of $x$. |
If $x$ is a real number, then $\Re(x^n) = x^n$. |
If $x$ is a real number, then $x^n$ is a real number. |
The real part of a natural number is the natural number itself. |
The imaginary part of a natural number is zero. |
The real part of an integer is the integer itself. |
The imaginary part of an integer is zero. |
The real part of a numeral is the numeral itself. |
The imaginary part of a numeral is zero. |
The real part of a real number is the number itself. |
The imaginary part of a real number is zero. |
The real part of a complex number divided by a numeral is the real part of the complex number divided by the numeral. |
The imaginary part of a complex number divided by a numeral is the imaginary part of the complex number divided by the numeral. |
The real part of a complex number divided by a natural number is equal to the real part of the complex number divided by the natural number. |
The imaginary part of a complex number divided by a natural number is equal to the imaginary part of the complex number divided by the natural number. |
If $z$ is a real number, then $\operatorname{Re}(z) = z$. |
The real part of the factorial of a natural number is the factorial of that natural number. |
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