Short Note

1. Laplace Transforms

Definition:

Elementary Transforms:

Function	Laplace Transform	Condition
		[cite_start] [cite: 58-60]
		[cite_start] [cite: 61-63]
(Dirac Delta)

Key Properties & Theorems:

• Linearity:
• First Shifting (Frequency):
• Second Shifting (Time):
• Time Scaling:
• Multiplication by :
• Division by :
• Periodic Functions (Period ):

Derivatives and Integrals:

• First Derivative:
• Second Derivative:
• Integral:

Convolution Theorem: Where

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2. Fourier Series

Standard Form (Period , Interval ):

General Period (Period , Interval ):

Half-Range Series (Interval ):

• Cosine Series (Even Ext.): , ,
• Sine Series (Odd Ext.): ,

Parseval’s Formula (Period ):

Complex Form (Period ):

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3. Partial Differential Equations (PDEs)

Classification (Second Order Linear PDE): [cite_start]Given [cite: 580]:

• Elliptic:
• Hyperbolic:
• Parabolic:

Key Equations:

• Laplace Equation (2D):
• Laplace (Polar):
• Heat Equation (1D):
• Wave Equation (1D):

Separation of Variables: Assume solution form:

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4. Series Solutions & Special Functions

Ordinary Point Solution:

Regular Singular Point (Frobenius Method):

• [cite_start]Requires solving the indicial equation for [cite: 904].

Bessel’s Equation:

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5. Fourier Transform

Definition:

Inverse Transform:

Key Properties:

• Linearity:
• Frequency Shifting:
• Time Shifting:
• Differentiation (Time):
• Differentiation (Frequency):
• Convolution:

Sine and Cosine Transforms ():

• Cosine Transform:
• Sine Transform:

Common Transforms:

• ()
• ()