Delving into the realm of Calculus and Analysis requires a solid understanding of the mathematical symbols that convey complex concepts. This comprehensive table serves as a valuable resource, offering a quick reference guide to symbols used in differentiation, integration, limits, series, and other fundamental aspects of mathematical analysis. Whether you're a student embarking on a calculus course or an enthusiast exploring advanced mathematical concepts, this guide will assist you in deciphering the symbolic language of calculus and analysis.
Symbol 
Symbol Name 
Meaning / definition 
Example 
\lim_{x\to x0}f(x) 
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 
\frac{dy}{dx} 
derivative 
derivative  Leibniz's notation 
d(3x3)/dx = 9x2 
\frac{d^2y}{dx^2} 
second derivative 
derivative of derivative 
d2(3x3)/dx2 = 18x 
\frac{d^ny}{dx^n} 
nth derivative 
n times derivation 

\dot{y} 
time derivative 
derivative by time  Newton's notation 

Dx y 
derivative 
derivative  Euler's notation 

Dx2y 
second derivative 
derivative of derivative 

\frac{\partial f(x,y)}{\partial x} 
partial derivative 
∂(x2+y2)/∂x = 2x 

∫ 
integral 
opposite to derivation 

∬ 
double integral 
integration of function of 2 variables 

∭ 
triple integral 
integration of function of 3 variables 

∮ 
closed contour / line integral 

∯ 
closed surface integral 

∰ 
closed volume integral 

[a,b] 
closed interval 
[a,b] = {x  a ≤ x ≤ b} 

(a,b) 
open interval 
(a,b) = {x  a < x < b} 

i 
imaginary unit 
i ≡ √1 
z = 3 + 2i 
z* 
complex conjugate 
z = a+bi → z*=abi 
z* = 3 + 2i 
z 
complex conjugate 
z = a+bi → z = abi 
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 

x * y 
convolution 
y(t) = x(t) * h(t) 

Laplace transform 
F(s) = {f (t)} 

Fourier transform 
X(ω) = {f (t)} 

δ 
delta function 

∞ 
lemniscate 
infinity symbol 
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