## Calculus & analysis symbols

SymbolSymbol NameMeaning / definitionExample
limit limit value of a function
ε epsilon represents a very small number, near zero ε  0
e e constant / Euler's number e = 2.718281828... e = lim (1+1/x)x , x→∞
y ' derivative derivative - Lagrange's notation (3x3)' = 9x2
y '' second derivative derivative of derivative (3x3)'' = 18x
y(n) nth derivative n times derivation (3x3)(3) = 18
derivative derivative - Leibniz's notation d(3x3)/dx = 9x2
second derivative derivative of derivative d2(3x3)/dx2 = 18x
nth derivative n times derivation
time derivative derivative by time - Newton's notation
time second derivative derivative of derivative
Dy derivative derivative - Euler's notation
Dx2y second derivative derivative of derivative
partial derivative   ∂(x2+y2)/∂x = 2x
integral opposite to derivation ∫ f(x)dx
∫∫ double integral integration of function of 2 variables ∫∫ f(x,y)dxdy
∫∫∫ triple integral integration of function of 3 variables ∫∫∫ f(x,y,z)dxdydz
closed contour / line integral
closed surface integral
closed volume integral
[a,b] closed interval [a,b] = {| a  x  b}
(a,b) open interval (a,b) = {| a < x < b}
i imaginary unit i ≡ √-1 z = 3 + 2i
z* complex conjugate = a+bi → z*=a-bi z* = 3 - 2i
z complex conjugate = a+bi → = a-bi z = 3 - 2i
Re(z) real part of a complex number z = a+bi → Re(z)=a Re(3 - 2i) = 3
Im(z) imaginary part of a complex number z = a+bi → Im(z)=b Im(3 - 2i) = -2
z | absolute value/magnitude of a complex number |z| = |a+bi| = √(a2+b2) |3 - 2i| = √13
arg(z) argument of a complex number The angle of the radius in the complex plane arg(3 + 2i) = 33.7°
nabla / del gradient / divergence operator f (x,y,z)
vector
unit vector
* y convolution y(t) = x(t) * h(t)
Laplace transform F(s) = {(t)}
Fourier transform X(ω) = {(t)}
δ delta function
lemniscate infinity symbol

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