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Mathematics

Real Analysis

Banach Spaces

Calculus of Variations

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Complex Analysis Fundamentals

Differentiation Rules

Laplace Transform

Ordinary Differential Equations

Partial Differential Equations

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Laplace Transform

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Exponential function $e^{at}$

\frac{1}{s-a}

Sine function $\sin(at)$

\frac{a}{s^2 + a^2}

Unit impulse at $t=a$ , $\delta(t-a)$

$e^{-as}$

Hyperbolic sine function $\sinh(at)$

\frac{a}{s^2 - a^2}

Rectangular pulse from $t=a$ to $t=b$ , $u(t-a) - u(t-b)$

\frac{e^{-as} - e^{-bs}}{s}

Ramp function $t$

\frac{1}{s^2}

Hyperbolic cosine function $\cosh(at)$

\frac{s}{s^2 - a^2}

Polynomial function $t^n$

\frac{n!}{s^{n+1}}

Sine function with damping $e^{-at}\sin(bt)$

\frac{b}{(s+a)^2 + b^2}

Natural logarithm $\ln(t)$

-\frac{1}{s} \ln(s) - \frac{\gamma}{s}

Delta function (Impulse function) $\delta(t)$

Cosine function $\cos(at)$

\frac{s}{s^2 + a^2}

Cosine function with damping $e^{-at}\cos(bt)$

\frac{s+a}{(s+a)^2 + b^2}

Exponential decay $e^{-at}$

\frac{1}{s+a}

Unit step function $u(t)$

\frac{1}{s}

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